kakz wrote:If the quantity 5^2 + 5^4 + 5^6 is written as (a + b)(a - b), in which both a and b are integers, which of the following could be the value of b?
(A)5
(B)10
(C)15
(D)20
(E)25
I have got this one in the manhattan site.... no oa given
edit: didn't see the posts above before writing this, but another approach:
First, if we are writing this number as a product (a+b)(a-b), then we're writing it as a product of two factors which differ by 2b. For example, if b were 5, then our product would be (a+5)(a-5), so we're just writing the number as a product of two factors which are 10 apart. So our task here is just to figure out how to write the number 5^2 + 5^4 + 5^6 as a product of two numbers which are, using the answer choices, 10 apart, 20 apart, 30 apart, 40 apart, or 50 apart. Notice also, because our factors differ by a multiple of 10, our two factors will need to have the same units digit.
Now we can factor 5^2 + 5^4 + 5^6 = 5^2(1 + 5^2 + 5^4) = 5^2 (1 + 25 + 625) = 5^2 * 651 = 3 * 5^2 * 7 * 31
So we know that a+b and a-b are made up of the primes 3, 5, 5, 7 and 31. They also must have the same units digit, so it cannot be that one of them is divisible by 5 (and thus end in 5, since all the primes are odd) and the other not (since then it would not end in 5). So each of a+b and a-b must be divisible by 5. Now that we know what to do with the two 5s, the only remaining primes to look at are 3, 7 and 31. Since our factors cannot be too far apart, given the answer choices, one of our factors will need to be 3*5*7 and the other 5*31. These differ by 50, so b, which is half the difference of our two factors, is 25.
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