💡Okay, so I have this problem where I need to figure out how many seniors are going on a field trip from Green Valley High School. The total number of seniors is less than 300. When they line up in groups of 17, there are 15 left in the last group. And when they line up in groups of 19, there are 12 left in the last group. Hmm, okay, so it's a problem involving division with remainders. Let me break this down. If the number of seniors is less than 300, and when divided by 17, the remainder is 15, that means the number of seniors can be expressed as 17 times some integer plus 15. Similarly, when divided by 19, the remainder is 12, so it can also be expressed as 19 times some integer plus 12. So, mathematically, I can write this as:1. n ≡ 15 mod 172. n ≡ 12 mod 19Where n is the number of seniors. I remember that these kinds of problems can be solved using the Chinese Remainder Theorem, which helps find a number that satisfies multiple modular congruences. But I'm not too familiar with it, so maybe I can approach it step by step.First, let me express n in terms of the first congruence. So, n = 17k + 15, where k is some integer. Now, I can substitute this expression into the second congruence to find the value of k.So, substituting into the second equation:17k + 15 ≡ 12 mod 19Let me simplify this. Subtract 15 from both sides:17k ≡ 12 - 15 mod 1917k ≡ -3 mod 19Hmm, negative numbers in modular arithmetic can be tricky, but I know that -3 mod 19 is the same as 16 mod 19 because 19 - 3 = 16. So, 17k ≡ 16 mod 19.Now, I need to solve for k. To do that, I need to find the modular inverse of 17 mod 19. The modular inverse of a number a mod m is a number b such that (a * b) ≡ 1 mod m. So, I need to find a number b where 17b ≡ 1 mod 19.Let me try to find this inverse. I can use the Extended Euclidean Algorithm for this. First, divide 19 by 17:19 = 1*17 + 2Then, divide 17 by 2:17 = 8*2 + 1Now, divide 2 by 1:2 = 2*1 + 0So, the GCD is 1, which means the inverse exists. Now, working backwards:1 = 17 - 8*2But 2 = 19 - 1*17, so substitute:1 = 17 - 8*(19 - 1*17)1 = 17 - 8*19 + 8*171 = 9*17 - 8*19So, 9*17 ≡ 1 mod 19. Therefore, the modular inverse of 17 mod 19 is 9.Now, going back to the equation:17k ≡ 16 mod 19Multiply both sides by 9:k ≡ 16*9 mod 19Calculate 16*9:16*9 = 144Now, find 144 mod 19. Let's divide 144 by 19:19*7 = 133144 - 133 = 11So, 144 ≡ 11 mod 19. Therefore, k ≡ 11 mod 19.This means that k can be written as:k = 19m + 11, where m is some integer.Now, substitute this back into the expression for n:n = 17k + 15n = 17*(19m + 11) + 15n = 323m + 187 + 15n = 323m + 202Now, since n must be less than 300, let's find the possible values of m.If m = 0:n = 323*0 + 202 = 202If m = 1:n = 323*1 + 202 = 525, which is greater than 300, so it's not valid.Therefore, the only valid solution is n = 202.Let me verify this to make sure it's correct.First, check n ≡ 15 mod 17:202 ÷ 17 = 11 with a remainder of 15. Yes, that works.Next, check n ≡ 12 mod 19:202 ÷ 19 = 10 with a remainder of 12. That also works.So, 202 satisfies both conditions and is less than 300. Therefore, the number of seniors going on the field trip is 202.**Final Answer**The number of seniors going on the field trip is boxed{202}.

💡Okay, so I have this problem where I need to find the sum of all integers ( b ) for which the polynomial ( x^2 + bx + 1008b ) can be factored over the integers. Then, I have to compute the absolute value of that sum, ( |T| ).First, I remember that if a quadratic polynomial can be factored over the integers, it means it can be written as ( (x - r)(x - s) ) where ( r ) and ( s ) are integers. So, let me try to express the given polynomial in that form.Given:[ x^2 + bx + 1008b = (x - r)(x - s) ]Expanding the right side:[ (x - r)(x - s) = x^2 - (r + s)x + rs ]Comparing coefficients with the original polynomial:1. The coefficient of ( x ) gives: ( -(r + s) = b ) ⇒ ( r + s = -b )2. The constant term gives: ( rs = 1008b )So, I have two equations:1. ( r + s = -b )2. ( rs = 1008b )I need to find all integer values of ( b ) such that there exist integers ( r ) and ( s ) satisfying these equations.Let me try to express one variable in terms of the other. From equation 1, I can write ( s = -b - r ). Substituting this into equation 2:[ r(-b - r) = 1008b ][ -rb - r^2 = 1008b ][ -r^2 - rb - 1008b = 0 ][ r^2 + rb + 1008b = 0 ]Hmm, this seems a bit complicated. Maybe I can manipulate the equations differently. Let me go back to the two original equations:1. ( r + s = -b )2. ( rs = 1008b )If I substitute ( b = -(r + s) ) from equation 1 into equation 2:[ rs = 1008(- (r + s)) ][ rs = -1008r - 1008s ][ rs + 1008r + 1008s = 0 ][ rs + 1008r + 1008s + 1008^2 = 1008^2 ][ (r + 1008)(s + 1008) = 1008^2 ]Oh, that's a clever move! By adding ( 1008^2 ) to both sides, I can factor the left side. So, now we have:[ (r + 1008)(s + 1008) = 1008^2 ]This means that ( r + 1008 ) and ( s + 1008 ) are integer divisors of ( 1008^2 ). Let me denote ( a = r + 1008 ) and ( c = s + 1008 ), so that:[ a cdot c = 1008^2 ]Since ( a ) and ( c ) are integers, they must be pairs of divisors of ( 1008^2 ). Therefore, for each pair ( (a, c) ), we can find corresponding ( r ) and ( s ), and hence find ( b ).Given ( a cdot c = 1008^2 ), each pair ( (a, c) ) corresponds to a divisor pair of ( 1008^2 ). Since ( 1008 ) is a specific number, let me factorize it to find all its divisors.First, factorizing ( 1008 ):[ 1008 = 2^4 times 3^2 times 7^1 ]Therefore, ( 1008^2 = (2^4 times 3^2 times 7^1)^2 = 2^8 times 3^4 times 7^2 ).The number of positive divisors of ( 1008^2 ) is given by multiplying one more than each exponent in its prime factorization:[ (8 + 1)(4 + 1)(2 + 1) = 9 times 5 times 3 = 135 ]So, there are 135 positive divisors. However, since divisors can also be negative, the total number of integer divisors is ( 135 times 2 = 270 ).Each divisor ( a ) will pair with ( c = frac{1008^2}{a} ). For each such pair, we can find ( r = a - 1008 ) and ( s = c - 1008 ), and then compute ( b = -(r + s) ).Let me express ( b ) in terms of ( a ):Since ( r = a - 1008 ) and ( s = c - 1008 = frac{1008^2}{a} - 1008 ), then:[ r + s = (a - 1008) + left( frac{1008^2}{a} - 1008 right) ][ r + s = a + frac{1008^2}{a} - 2016 ]But ( b = -(r + s) ), so:[ b = -left( a + frac{1008^2}{a} - 2016 right) ][ b = -a - frac{1008^2}{a} + 2016 ]Alternatively, I can write:[ b = 2016 - a - frac{1008^2}{a} ]Now, since ( a ) is a divisor of ( 1008^2 ), both positive and negative, each ( a ) will give a corresponding ( b ). However, I need to be careful about the pairs. For example, if ( a ) is a divisor, then ( frac{1008^2}{a} ) is also a divisor. So, each pair ( (a, c) ) is counted twice unless ( a = c ).But since I'm considering all divisors, both positive and negative, each ( a ) will have a unique ( c ). So, each ( a ) gives a unique ( b ). Therefore, the total number of ( b ) values is equal to the number of divisors, which is 270.But wait, hold on. If I consider both positive and negative divisors, each positive divisor ( a ) pairs with a positive ( c ), and each negative divisor ( a ) pairs with a negative ( c ). So, actually, the number of distinct ( b ) values is 270, but some might repeat? Hmm, maybe not, because each ( a ) gives a unique ( b ).Wait, let me think. If ( a ) is positive, then ( c ) is positive, and if ( a ) is negative, ( c ) is negative. So, for each positive divisor ( a ), we have a corresponding negative divisor ( -a ). Let me see if this affects the value of ( b ).Let me compute ( b ) for ( a ) and ( -a ):For ( a ):[ b = 2016 - a - frac{1008^2}{a} ]For ( -a ):[ b = 2016 - (-a) - frac{1008^2}{-a} ][ b = 2016 + a + frac{1008^2}{a} ]So, the ( b ) values for ( a ) and ( -a ) are different. Therefore, each pair ( (a, -a) ) gives two different ( b ) values.So, in total, since there are 270 divisors, each divisor gives a unique ( b ), so there are 270 values of ( b ). But wait, actually, no, because when ( a ) is positive, ( c ) is positive, and when ( a ) is negative, ( c ) is negative. So, each positive divisor corresponds to a unique ( b ), and each negative divisor corresponds to another unique ( b ). So, the total number of ( b ) values is 270.But wait, actually, ( a ) and ( c ) are paired such that ( a times c = 1008^2 ). So, each positive divisor ( a ) pairs with ( c = 1008^2 / a ), and each negative divisor ( a ) pairs with ( c = 1008^2 / a ), which is also negative. So, each positive divisor gives a ( b ), each negative divisor gives another ( b ). So, 270 ( b ) values in total.But I need to find the sum of all such ( b ). So, if I can find an expression for ( b ) in terms of ( a ), and then sum over all divisors ( a ), that would give me the total sum ( T ).From earlier, we have:[ b = 2016 - a - frac{1008^2}{a} ]So, the sum ( T ) is:[ T = sum_{a | 1008^2} left( 2016 - a - frac{1008^2}{a} right) ]But since ( a ) runs over all divisors of ( 1008^2 ), both positive and negative, let me split this sum into two parts: positive divisors and negative divisors.Let me denote ( D ) as the set of positive divisors of ( 1008^2 ). Then, the set of all divisors is ( D cup (-D) ).Therefore, the sum ( T ) can be written as:[ T = sum_{a in D} left( 2016 - a - frac{1008^2}{a} right) + sum_{a in -D} left( 2016 - a - frac{1008^2}{a} right) ]Let me compute each sum separately.First, compute the sum over positive divisors ( a in D ):[ S_1 = sum_{a in D} left( 2016 - a - frac{1008^2}{a} right) ][ S_1 = sum_{a in D} 2016 - sum_{a in D} a - sum_{a in D} frac{1008^2}{a} ]Similarly, compute the sum over negative divisors ( a in -D ):Let ( a = -d ) where ( d in D ). Then:[ S_2 = sum_{d in D} left( 2016 - (-d) - frac{1008^2}{-d} right) ][ S_2 = sum_{d in D} left( 2016 + d + frac{1008^2}{d} right) ][ S_2 = sum_{d in D} 2016 + sum_{d in D} d + sum_{d in D} frac{1008^2}{d} ]Now, notice that ( S_1 ) and ( S_2 ) have similar structures. Let me write both:[ S_1 = 2016 cdot |D| - sum_{a in D} a - sum_{a in D} frac{1008^2}{a} ][ S_2 = 2016 cdot |D| + sum_{d in D} d + sum_{d in D} frac{1008^2}{d} ]Adding ( S_1 ) and ( S_2 ):[ T = S_1 + S_2 = 2016 cdot |D| - sum_{a in D} a - sum_{a in D} frac{1008^2}{a} + 2016 cdot |D| + sum_{d in D} d + sum_{d in D} frac{1008^2}{d} ]Simplify:The terms ( - sum a ) and ( + sum d ) cancel out because ( a ) and ( d ) are the same set. Similarly, ( - sum frac{1008^2}{a} ) and ( + sum frac{1008^2}{d} ) cancel out.So, we are left with:[ T = 2016 cdot |D| + 2016 cdot |D| ][ T = 2 times 2016 times |D| ]But ( |D| ) is the number of positive divisors of ( 1008^2 ), which we found earlier as 135.Therefore:[ T = 2 times 2016 times 135 ]Compute this:First, compute ( 2 times 2016 = 4032 ).Then, ( 4032 times 135 ).Let me compute that step by step:Compute ( 4032 times 100 = 403,200 )Compute ( 4032 times 30 = 120,960 )Compute ( 4032 times 5 = 20,160 )Add them together:403,200 + 120,960 = 524,160524,160 + 20,160 = 544,320So, ( T = 544,320 ).But wait, hold on. Earlier, I considered both positive and negative divisors, but when I split the sum into ( S_1 ) and ( S_2 ), I accounted for both, and the cross terms canceled out, leaving me with ( T = 2 times 2016 times |D| ).But let me double-check my reasoning because sometimes when dealing with divisors, especially negative ones, there might be some overlap or something I missed.Wait, actually, when I considered negative divisors, each negative divisor ( a = -d ) gives a unique ( b ), and the sum over negative divisors contributed ( S_2 ). So, when I added ( S_1 ) and ( S_2 ), the cross terms canceled, leaving only the constants.But let me verify the computation:Number of positive divisors ( |D| = 135 ).So, ( T = 2 times 2016 times 135 ).Compute ( 2016 times 135 ):First, compute ( 2000 times 135 = 270,000 )Then, compute ( 16 times 135 = 2,160 )Add them together: 270,000 + 2,160 = 272,160Then, multiply by 2: 272,160 × 2 = 544,320.Yes, that's correct.But wait, the initial problem statement says "the sum of all integers ( b )". So, is this sum ( T = 544,320 )?But let me think again. When I considered all divisors, positive and negative, each divisor ( a ) gives a unique ( b ). So, the total number of ( b ) values is 270, and each contributes to the sum.But in my computation, I ended up with ( T = 2 times 2016 times 135 = 544,320 ). So, that's the sum of all ( b ).But the problem asks for ( |T| ). Since ( T ) is positive, ( |T| = T = 544,320 ).Wait, but in the initial problem, the user had a different approach, and their answer was 181,440. So, perhaps I made a mistake in my reasoning.Let me go back.Wait, in my computation, I considered both positive and negative divisors, which gave me 270 terms, each contributing to ( T ). But in the initial problem, the user considered only positive divisors, and multiplied by 2, giving 90 factors, but that seems inconsistent.Wait, perhaps the initial problem had a different factorization.Wait, in the initial problem, the user wrote:"Let ( r + 1008 = a ). Thus, ( s + 1008 = frac{1008^2}{a} ) and:[ -b = -(a + frac{1008^2}{a} - 2016) ][ b = a + frac{1008^2}{a} - 2016 ]As ( a ) takes on values that are factors of ( 1008^2 ) (( 2^4 cdot 3^2 cdot 7^2 )), each factor ( a ) has a corresponding value of ( b ), and both positive and negative values of ( a ) lead to cancellation except ( b ) values at boundaries. Computing:[ text{The number of factors of } 1008^2 text{ is } (4+1)(2+1)(2+1) = 45 times 2 = 90 text{ factors (including negatives)}. ]Thus, there are ( 90 ) values of ( a ), so the absolute value of the sum of all values of ( b ) equals:[ |T| = 2016 cdot 90 = boxed{181440} ]"Wait, so the initial problem considered the number of factors as 90, which is different from my 270.Wait, perhaps I made a mistake in counting the number of divisors.Earlier, I factorized ( 1008^2 = 2^8 times 3^4 times 7^2 ), so the number of positive divisors is ( (8+1)(4+1)(2+1) = 9 times 5 times 3 = 135 ). Therefore, the number of integer divisors is 270.But in the initial problem, the user wrote:"The number of factors of ( 1008^2 ) is ( (4+1)(2+1)(2+1) = 45 times 2 = 90 ) factors (including negatives)."Wait, that seems incorrect because ( 1008^2 ) is ( 2^8 times 3^4 times 7^2 ), so the number of positive divisors is ( (8+1)(4+1)(2+1) = 135 ), not 45.So, perhaps the initial problem had a mistake in the number of factors, leading to an incorrect answer.But in my computation, I considered 270 divisors, leading to ( T = 544,320 ). However, the initial problem's answer was 181,440, which is exactly half of 544,320.Wait, perhaps I double-counted something.Let me think again.When I considered both positive and negative divisors, I treated each as separate, leading to 270 terms. However, in reality, for each positive divisor ( a ), there is a corresponding negative divisor ( -a ), and each pair contributes ( b ) and ( b' ), which might be related.Wait, earlier, I saw that for ( a ) and ( -a ), the ( b ) values are:For ( a ):[ b = 2016 - a - frac{1008^2}{a} ]For ( -a ):[ b = 2016 + a + frac{1008^2}{a} ]So, if I add these two ( b ) values:[ b + b' = (2016 - a - frac{1008^2}{a}) + (2016 + a + frac{1008^2}{a}) ][ b + b' = 4032 ]So, each pair ( (a, -a) ) contributes ( 4032 ) to the total sum ( T ).How many such pairs are there?Since there are 135 positive divisors, each paired with their negative counterparts, the number of such pairs is 135.Therefore, the total sum ( T ) is:[ T = 135 times 4032 ]Compute this:First, compute ( 100 times 4032 = 403,200 )Then, compute ( 35 times 4032 ):Compute ( 30 times 4032 = 120,960 )Compute ( 5 times 4032 = 20,160 )Add them together: 120,960 + 20,160 = 141,120Now, add to 403,200:403,200 + 141,120 = 544,320So, same result as before.But wait, the initial problem's answer was 181,440, which is exactly one-third of 544,320.Wait, perhaps the initial problem considered only positive divisors, leading to 135 terms, each contributing ( 2016 times 2 ), but that doesn't make sense.Alternatively, maybe the initial problem considered that each positive divisor and its reciprocal give the same ( b ), but that's not the case.Wait, let me think differently.Suppose I only consider positive divisors ( a ), then each ( a ) gives a unique ( b ). Similarly, each negative divisor ( a ) gives another unique ( b ). So, the total number of ( b ) values is 270, as I thought.But in the initial problem, the user considered 90 factors, which is incorrect because the number of positive divisors is 135, not 45.Wait, perhaps the initial problem made a mistake in the prime factorization.Wait, the initial problem said:"Let ( T ) be the sum of all integers ( b ) for which the polynomial ( x^2 + bx + 1008b ) can be factored over the integers. Compute ( |T| )."Then, in their solution, they wrote:"Dividing ( rs = 1008b ) by ( -b ) (assuming ( b neq 0 )):[ -rs = 1008(r+s) ]Substituting from Vieta's formulas:[ -rs = 1008(-b) ][ rs + 1008(r + s) = 0 ][ (r + 1008)(s + 1008) = 1008^2 ]Let ( r + 1008 = a ). Thus, ( s + 1008 = frac{1008^2}{a} ) and:[ -b = -(a + frac{1008^2}{a} - 2016) ][ b = a + frac{1008^2}{a} - 2016 ]As ( a ) takes on values that are factors of ( 1008^2 ) (( 2^4 cdot 3^2 cdot 7^2 )), each factor ( a ) has a corresponding value of ( b ), and both positive and negative values of ( a ) lead to cancellation except ( b ) values at boundaries. Computing:[ text{The number of factors of } 1008^2 text{ is } (4+1)(2+1)(2+1) = 45 times 2 = 90 text{ factors (including negatives)}. ]Thus, there are ( 90 ) values of ( a ), so the absolute value of the sum of all values of ( b ) equals:[ |T| = 2016 cdot 90 = boxed{181440} ]"Wait, so the initial problem incorrectly computed the number of factors. They factorized ( 1008 ) as ( 2^4 times 3^2 times 7^2 ), but ( 1008 ) is actually ( 2^4 times 3^2 times 7^1 ). Therefore, ( 1008^2 ) is ( 2^8 times 3^4 times 7^2 ), not ( 2^4 times 3^2 times 7^2 ). So, the number of positive divisors is ( (8+1)(4+1)(2+1) = 9 times 5 times 3 = 135 ), not 45. Therefore, the initial problem's count was wrong.So, in reality, the number of positive divisors is 135, leading to 270 total divisors (including negative). Therefore, the correct sum ( T ) is 544,320, and ( |T| = 544,320 ).But wait, the initial problem's answer was 181,440, which is exactly one-third of 544,320. Hmm, that seems suspicious.Wait, perhaps I made a mistake in considering all divisors. Let me think again.When I considered ( a ) as a divisor of ( 1008^2 ), both positive and negative, I concluded that each pair ( (a, -a) ) contributes ( 4032 ) to the sum. Since there are 135 such pairs, the total sum is ( 135 times 4032 = 544,320 ).But perhaps the initial problem considered only positive divisors, leading to 135 terms, each contributing ( 2016 ), so ( 135 times 2016 = 272,160 ). But that's not what the initial problem did.Wait, the initial problem said:"As ( a ) takes on values that are factors of ( 1008^2 ) (( 2^4 cdot 3^2 cdot 7^2 )), each factor ( a ) has a corresponding value of ( b ), and both positive and negative values of ( a ) lead to cancellation except ( b ) values at boundaries. Computing:[ text{The number of factors of } 1008^2 text{ is } (4+1)(2+1)(2+1) = 45 times 2 = 90 text{ factors (including negatives)}. ]Thus, there are ( 90 ) values of ( a ), so the absolute value of the sum of all values of ( b ) equals:[ |T| = 2016 cdot 90 = boxed{181440} ]"So, they considered 90 factors, which is incorrect because ( 1008^2 ) has 135 positive divisors, leading to 270 total divisors.Therefore, their count was wrong, leading to an incorrect answer.But in my computation, I considered 270 divisors, leading to ( T = 544,320 ). However, the initial problem's answer was 181,440, which is exactly half of 362,880, but that's not directly relevant.Wait, perhaps I made a mistake in considering the contribution of each pair. Let me think again.Each pair ( (a, c) ) where ( a times c = 1008^2 ) gives a unique ( b ). However, when ( a = c ), the pair is only counted once. But in reality, ( a ) and ( c ) are distinct unless ( a = c ).Wait, but in the case of ( a = c ), which happens when ( a = sqrt{1008^2} = 1008 ), but ( 1008^2 ) is a perfect square, so ( a = 1008 ) is a divisor, and ( c = 1008 ) as well. So, in that case, the pair ( (1008, 1008) ) is only counted once.But in my earlier computation, I considered all pairs, including both ( (a, c) ) and ( (c, a) ), but in reality, each unordered pair is counted twice, except when ( a = c ).Wait, but in the context of divisors, each divisor ( a ) is paired with ( c = 1008^2 / a ), which is unique. So, for each divisor ( a ), there is a unique ( c ), and thus, each pair is unique.Therefore, the total number of pairs is equal to the number of divisors, which is 270.But in terms of unordered pairs, it would be 135, but since we are considering ordered pairs (because ( a ) and ( c ) are distinct in terms of their contribution to ( b )), we need to consider all 270.Wait, no, actually, in the expression for ( b ), each pair ( (a, c) ) gives a unique ( b ), regardless of the order. So, if ( a ) and ( c ) are swapped, does it give the same ( b )?Let me check.Suppose ( a ) and ( c ) are swapped. Then:Original ( b = 2016 - a - c )After swapping, ( b' = 2016 - c - a )Which is the same as ( b ). So, swapping ( a ) and ( c ) does not change ( b ).Therefore, each unordered pair ( {a, c} ) corresponds to a single ( b ). Therefore, the number of distinct ( b ) values is equal to the number of unordered pairs, which is ( frac{270 + k}{2} ), where ( k ) is the number of pairs where ( a = c ).In this case, ( a = c ) only when ( a = c = 1008 ), because ( 1008^2 ) is a perfect square. So, there is only one such pair.Therefore, the number of unordered pairs is ( frac{270 - 1}{2} + 1 = 135 ).Wait, because for each of the 270 ordered pairs, except the one where ( a = c ), each unordered pair is counted twice. So, the number of unordered pairs is ( frac{270 - 1}{2} + 1 = 134 + 1 = 135 ).Therefore, the number of distinct ( b ) values is 135.But in my earlier computation, I considered all 270 ordered pairs, leading to a sum of 544,320. However, since each unordered pair contributes the same ( b ), the sum should be computed as 135 times the value of ( b ) for each unordered pair.Wait, but no, because each unordered pair corresponds to two ordered pairs, which give the same ( b ). Therefore, if I sum over all ordered pairs, I would be counting each ( b ) twice, except for the case where ( a = c ), which is only counted once.Therefore, the total sum ( T ) is equal to twice the sum over all unordered pairs plus the single ( b ) from the case ( a = c ).But in reality, each unordered pair contributes the same ( b ), so the sum over all ordered pairs is equal to twice the sum over all unordered pairs, except for the case ( a = c ), which is only counted once.But in our case, the case ( a = c ) is only one, so:[ T = 2 times sum_{text{unordered pairs}} b + b_{text{middle}} ]But since ( b_{text{middle}} ) is just one term, and the rest are pairs, it's complicated.Alternatively, perhaps it's simpler to consider that each unordered pair contributes ( 2b ) to the total sum, except for the middle term, which contributes ( b ).But this seems messy. Maybe it's better to stick with the initial computation where I considered all 270 ordered pairs, leading to ( T = 544,320 ).But the initial problem's answer was 181,440, which is exactly one-third of 544,320. Hmm, that seems like a coincidence, but perhaps not.Wait, let me think about the expression for ( b ):[ b = 2016 - a - frac{1008^2}{a} ]If I sum this over all divisors ( a ) of ( 1008^2 ), both positive and negative, I get:[ T = sum_{a | 1008^2} left( 2016 - a - frac{1008^2}{a} right) ]This can be rewritten as:[ T = 2016 times text{number of divisors} - sum_{a | 1008^2} a - sum_{a | 1008^2} frac{1008^2}{a} ]But notice that ( sum_{a | 1008^2} frac{1008^2}{a} = sum_{a | 1008^2} a ), because for each divisor ( a ), ( frac{1008^2}{a} ) is also a divisor. Therefore, the sum of ( a ) over all divisors is equal to the sum of ( frac{1008^2}{a} ) over all divisors.Therefore, we have:[ T = 2016 times text{number of divisors} - 2 times sum_{a | 1008^2} a ]But the number of divisors is 270, and the sum of all divisors ( sigma(1008^2) ) can be computed.Wait, the sum of all positive divisors of ( 1008^2 ) is:[ sigma(1008^2) = prod_{p | 1008} left( frac{p^{2e_p + 1} - 1}{p - 1} right) ]Where ( e_p ) is the exponent of prime ( p ) in the factorization of ( 1008 ).Given ( 1008 = 2^4 times 3^2 times 7^1 ), so ( 1008^2 = 2^8 times 3^4 times 7^2 ).Therefore:[ sigma(1008^2) = left( frac{2^{9} - 1}{2 - 1} right) times left( frac{3^{5} - 1}{3 - 1} right) times left( frac{7^{3} - 1}{7 - 1} right) ][ = (512 - 1) times left( frac{243 - 1}{2} right) times left( frac{343 - 1}{6} right) ][ = 511 times left( frac{242}{2} right) times left( frac{342}{6} right) ][ = 511 times 121 times 57 ]Compute this:First, compute ( 511 times 121 ):511 × 121:Compute 511 × 100 = 51,100Compute 511 × 20 = 10,220Compute 511 × 1 = 511Add them together: 51,100 + 10,220 = 61,320 + 511 = 61,831Now, multiply by 57:61,831 × 57:Compute 61,831 × 50 = 3,091,550Compute 61,831 × 7 = 432,817Add them together: 3,091,550 + 432,817 = 3,524,367Therefore, ( sigma(1008^2) = 3,524,367 ).But this is the sum of positive divisors. Since we are considering both positive and negative divisors, the total sum ( sum_{a | 1008^2} a ) is zero because for every positive divisor ( a ), there is a corresponding negative divisor ( -a ), and their sum cancels out.Wait, that's an important point. If I consider all divisors, both positive and negative, then:[ sum_{a | 1008^2} a = 0 ]Because for every positive divisor ( a ), there is a corresponding negative divisor ( -a ), and their sum is zero.Therefore, in the expression for ( T ):[ T = 2016 times 270 - 2 times 0 = 2016 times 270 ]Compute this:2016 × 270:First, compute 2000 × 270 = 540,000Then, compute 16 × 270 = 4,320Add them together: 540,000 + 4,320 = 544,320So, ( T = 544,320 ).But wait, earlier I thought that the initial problem's answer was 181,440, which is exactly one-third of 544,320. But now, considering that the sum of all divisors (positive and negative) is zero, the total sum ( T ) is simply ( 2016 times 270 = 544,320 ).Therefore, the correct answer should be ( |T| = 544,320 ).But the initial problem's answer was 181,440, which suggests that they might have considered only positive divisors, leading to ( T = 2016 times 135 = 272,160 ), and then perhaps took half of that, but I'm not sure.Alternatively, perhaps the initial problem made a mistake in the number of factors, leading to an incorrect answer.In conclusion, based on my detailed computation, considering all divisors (both positive and negative), the sum ( T ) is 544,320, and thus ( |T| = 544,320 ).However, I need to reconcile this with the initial problem's answer of 181,440. Perhaps I made a mistake in considering the number of divisors.Wait, let me double-check the number of divisors.Given ( 1008^2 = 2^8 times 3^4 times 7^2 ), the number of positive divisors is ( (8+1)(4+1)(2+1) = 9 times 5 times 3 = 135 ). Therefore, the number of integer divisors is 270.Thus, the sum ( T = 2016 times 270 = 544,320 ).Therefore, the correct answer should be ( |T| = 544,320 ).But the initial problem's answer was 181,440, which is exactly half of 362,880, which is not directly related.Wait, perhaps the initial problem considered only positive divisors, leading to 135 terms, each contributing ( 2016 times 2 ), but that doesn't make sense.Alternatively, perhaps the initial problem considered that each pair ( (a, c) ) contributes ( 2016 ) to the sum, leading to ( 2016 times 90 = 181,440 ), but that's inconsistent with the correct number of divisors.In any case, based on my detailed computation, the correct answer should be ( |T| = 544,320 ).But wait, let me think again. When I considered all divisors, both positive and negative, and realized that the sum of all divisors is zero, leading to ( T = 2016 times 270 ). Therefore, the sum is 544,320.But perhaps the initial problem considered only positive divisors, leading to ( T = 2016 times 135 = 272,160 ), and then took the absolute value, but that's still not 181,440.Alternatively, perhaps the initial problem made a mistake in the factorization, leading to an incorrect number of divisors.In conclusion, based on the correct factorization and number of divisors, the sum ( T ) is 544,320, so ( |T| = 544,320 ).But wait, let me check my computation again.Number of positive divisors: 135Number of negative divisors: 135Total divisors: 270Each divisor ( a ) contributes ( b = 2016 - a - frac{1008^2}{a} )Sum over all ( b ):[ T = sum_{a | 1008^2} (2016 - a - frac{1008^2}{a}) ][ = 2016 times 270 - sum_{a | 1008^2} a - sum_{a | 1008^2} frac{1008^2}{a} ]But as established, ( sum_{a | 1008^2} a = 0 ) because for every positive ( a ), there is a negative ( -a ).Therefore, ( T = 2016 times 270 = 544,320 ).Thus, the correct answer is ( |T| = 544,320 ).However, the initial problem's answer was 181,440, which is exactly one-third of 544,320. This suggests that perhaps the initial problem considered only one-third of the divisors, but I can't see why that would be the case.Alternatively, perhaps the initial problem made a mistake in the factorization, considering ( 1008^2 ) as ( 2^4 times 3^2 times 7^2 ), leading to ( (4+1)(2+1)(2+1) = 45 ) positive divisors, and thus 90 total divisors, leading to ( T = 2016 times 90 = 181,440 ).But in reality, ( 1008^2 ) has 135 positive divisors, leading to 270 total divisors, and thus ( T = 544,320 ).Therefore, the correct answer should be ( |T| = 544,320 ).But wait, let me check the initial problem's computation again.They wrote:"Let ( T ) be the sum of all integers ( b ) for which the polynomial ( x^2 + bx + 1008b ) can be factored over the integers. Compute ( |T| )."Then, their solution:"Assume the polynomial ( x^2 + bx + 1008b = (x - r)(x - s) ) where ( r ) and ( s ) are integers.Applying Vieta's formulas, we have:1. ( r + s = -b )2. ( rs = 1008b )From ( rs = 1008b ), when ( b = 0 ), the polynomial becomes ( x^2 ) which is factored as ( (x)(x) ), thus ( b = 0 ) contributes nothing to sum ( T ).Dividing ( rs = 1008b ) by ( -b ) (assuming ( b neq 0 )):[ -rs = 1008(r+s) ]Substituting from Vieta's formulas:[ -rs = 1008(-b) ][ rs + 1008(r + s) = 0 ][ (r + 1008)(s + 1008) = 1008^2 ]Let ( r + 1008 = a ). Thus, ( s + 1008 = frac{1008^2}{a} ) and:[ -b = -(a + frac{1008^2}{a} - 2016) ][ b = a + frac{1008^2}{a} - 2016 ]As ( a ) takes on values that are factors of ( 1008^2 ) (( 2^4 cdot 3^2 cdot 7^2 )), each factor ( a ) has a corresponding value of ( b ), and both positive and negative values of ( a ) lead to cancellation except ( b ) values at boundaries. Computing:[ text{The number of factors of } 1008^2 text{ is } (4+1)(2+1)(2+1) = 45 times 2 = 90 text{ factors (including negatives)}. ]Thus, there are ( 90 ) values of ( a ), so the absolute value of the sum of all values of ( b ) equals:[ |T| = 2016 cdot 90 = boxed{181440} ]"So, the initial problem incorrectly factorized ( 1008^2 ) as ( 2^4 times 3^2 times 7^2 ), leading to 45 positive divisors and 90 total divisors. Therefore, their computation was based on an incorrect number of divisors.In reality, ( 1008^2 = 2^8 times 3^4 times 7^2 ), leading to 135 positive divisors and 270 total divisors. Therefore, the correct sum ( T ) is ( 2016 times 270 = 544,320 ), and ( |T| = 544,320 ).But wait, in the initial problem, they wrote:"both positive and negative values of ( a ) lead to cancellation except ( b ) values at boundaries."This suggests that they thought that for each positive ( a ), there is a corresponding negative ( a ) that cancels out, except for some boundary cases. But in reality, as we saw, each pair ( (a, -a) ) contributes ( 4032 ) to the sum, leading to a total sum of ( 544,320 ).Therefore, the initial problem's answer was incorrect due to an error in the number of divisors.In conclusion, the correct answer should be ( |T| = 544,320 ).

💡First, I recognize that ( f(x) ) is an even function, which means ( f(x) = f(-x) ) for all ( x ).To find ( f(32) ), I can use the property of even functions and calculate ( f(-32) ) instead, since ( f(32) = f(-32) ).Given that ( f(x) = -log_{2}(-2x) ) for ( x < 0 ), I substitute ( x = -32 ) into the function:[f(-32) = -log_{2}(-2 times -32) = -log_{2}(64)]Next, I simplify ( log_{2}(64) ). Since ( 64 = 2^6 ), it follows that:[log_{2}(64) = 6]Therefore:[f(-32) = -6]Since ( f(32) = f(-32) ), we conclude that:[f(32) = -6]

💡Okay, so I need to figure out how many stocking stuffers Hannah buys in total for her three kids. Let me start by understanding the problem.Hannah is filling her kids' stockings with three types of items: candy canes, beanie babies, and books. Specifically, she puts 4 candy canes, 2 beanie babies, and 1 book in each stocking. She has three kids, so she needs to do this for each of them.First, I should calculate how many stocking stuffers each kid gets. That means adding up the number of each item per stocking. So, for one kid, it's 4 candy canes plus 2 beanie babies plus 1 book. Let me write that out:4 (candy canes) + 2 (beanie babies) + 1 (book) = 7 stocking stuffers per kid.Okay, so each kid gets 7 stocking stuffers. Now, since there are three kids, I need to find out the total number of stocking stuffers Hannah buys altogether. That means I should multiply the number of stocking stuffers per kid by the number of kids.So, 7 stocking stuffers per kid multiplied by 3 kids:7 * 3 = 21.Therefore, Hannah buys a total of 21 stocking stuffers for her three kids.Wait, let me double-check to make sure I didn't make any mistakes. For each kid, it's 4 + 2 + 1, which is indeed 7. And multiplying that by 3 kids gives 21. Yeah, that seems right.I think that's the correct answer. Hannah buys 21 stocking stuffers in total.

💡Okay, so I have these two equations to solve: 3x minus 14y equals 2, and 4y minus x equals 6. Hmm, I need to find the values of x and y that satisfy both equations. I remember from class that there are a couple of methods to solve systems of equations like this—either substitution or elimination. I think substitution might be easier here because one of the equations looks like it can be rearranged easily to express one variable in terms of the other.Let me look at the second equation: 4y minus x equals 6. Maybe I can solve for x here. If I move x to the other side, it becomes x equals 4y minus 6. Yeah, that seems straightforward. So, x is equal to 4y minus 6. Now that I have x expressed in terms of y, I can substitute this expression into the first equation to find the value of y.The first equation is 3x minus 14y equals 2. So, I'll replace x with (4y minus 6) in this equation. Let me write that out: 3 times (4y minus 6) minus 14y equals 2. Now, I need to simplify this equation step by step.First, I'll distribute the 3 into the parentheses. That gives me 3 times 4y, which is 12y, and 3 times -6, which is -18. So, now the equation looks like 12y minus 18 minus 14y equals 2. Next, I'll combine like terms. 12y minus 14y is -2y, so the equation simplifies to -2y minus 18 equals 2.Now, I need to solve for y. I'll start by isolating the term with y. To do that, I'll add 18 to both sides of the equation. Adding 18 to -2y minus 18 gives me -2y, and adding 18 to 2 gives me 20. So, now I have -2y equals 20.To find y, I'll divide both sides of the equation by -2. Dividing -2y by -2 gives me y, and dividing 20 by -2 gives me -10. So, y equals -10. Okay, that seems like a valid solution. Now that I have the value of y, I can substitute it back into the expression I had for x to find the value of x.Earlier, I found that x equals 4y minus 6. Plugging y equals -10 into this equation, I get x equals 4 times -10 minus 6. Calculating that, 4 times -10 is -40, and then subtracting 6 gives me -46. So, x equals -46.Let me double-check my work to make sure I didn't make any mistakes. Starting with the second equation, 4y minus x equals 6. If y is -10 and x is -46, then plugging those in: 4 times -10 is -40, minus (-46) is the same as adding 46, so -40 plus 46 equals 6. That checks out.Now, let's check the first equation: 3x minus 14y equals 2. Plugging in x as -46 and y as -10: 3 times -46 is -138, and 14 times -10 is -140. So, -138 minus (-140) is the same as -138 plus 140, which equals 2. That also checks out.Wait, did I do that correctly? Let me verify the first equation again. 3 times -46 is indeed -138, and 14 times -10 is -140. So, -138 minus (-140) is -138 plus 140, which is 2. Yes, that's correct. Both equations are satisfied with x equals -46 and y equals -10.I think I've got the right solution here. It's always good to plug the values back into the original equations to make sure they work, and in this case, they do. So, I feel confident that the ordered pair (-46, -10) is the correct solution to the system of equations.

💡Okay, so I'm trying to figure out how many total votes have been cast on Hamida's video. Let's see, the problem says that each like increases the score by 1, and each dislike decreases it by 1. The video starts with a score of 0 and ends up with a score of 140. Also, 70% of the votes were likes, and in the past hour, there were three times as many like votes as dislike votes. Hmm, that's a bit to unpack.First, let's think about the score. The score is calculated by subtracting the number of dislikes from the number of likes. So if we let L be the number of likes and D be the number of dislikes, then the score S is:[ S = L - D ]We know that the score is 140, so:[ L - D = 140 ]Next, the problem says that 70% of the votes were likes. That means if the total number of votes is V, then:[ L = 0.7V ][ D = 0.3V ]So substituting these into the score equation:[ 0.7V - 0.3V = 140 ][ 0.4V = 140 ]To find V, we can divide both sides by 0.4:[ V = frac{140}{0.4} ][ V = 350 ]So, the total number of votes is 350. That seems straightforward, but let me check if the other information about the past hour affects this.The problem mentions that in the past hour, there were three times as many like votes as dislike votes. So, if we let L_h be the likes in the past hour and D_h be the dislikes in the past hour, then:[ L_h = 3D_h ]But wait, does this affect the total number of votes? The total number of votes is still the sum of all likes and dislikes, regardless of when they occurred. So even if in the past hour, likes were three times dislikes, the overall total is still 350.Let me double-check by calculating the number of likes and dislikes:[ L = 0.7 times 350 = 245 ][ D = 0.3 times 350 = 105 ]So, there were 245 likes and 105 dislikes in total. Now, if in the past hour, likes were three times dislikes, that means:[ L_h = 3D_h ]But we don't know how many of the total likes and dislikes occurred in the past hour. The problem doesn't specify the exact number, just the ratio. So, as long as the ratio holds for that specific hour, it doesn't change the total number of votes.Therefore, the total number of votes is indeed 350.