Which one of the following numbers can not be expressed as the difference of the squares of two integers?

Question

a) 314159265
b) 314159266
c) 314159267
d) 314159268
e) 314159269

Answer 1

b)

Note that 314159266 is of the form 4k + 2, where k is an integer. Since any integer is of the form 2n or 2n +1, this means that all squares of integers are either of the form 4m or 4m + 1. No matter how we try to do the subtraction of two numbers of the form 4m or 4m + 1, we won't get a number of the form 4k + 2:

4m - 4n = 4k
4m - (4n + 1) = 4k + 3
(4m + 1) - 4n = 4k + 1
(4m + 1) - (4n + 1) = 4k.

Cheers!

Answer 2

The posters selecting b are correct. I'll just offer another way to solve this:

Factor x^2 - y^2 to (x+y)(x-y). Now note that:

If x and y are both odd or both even, then (x+y) and (x-y) are both even. Therefore (x+y)(x-y) has to be a multiple of 4.

If either x or y, but not both, is even (i.e. one is even, the other is odd), then (x+y) and (x-y) are both odd. Therefore (x+y)(x-y) has to be odd.

Therefore, the difference of two squares must be either odd or a multiple of 4 - it can't be even but not a multiple of 4 (2, 6, 10, etc.). To determine if a number is even but not a multiple of 4, you only have to check the last two digits, so the number ending in 66 is the correct answer.

By the way, you can also prove that ALL odd numbers and ALL multiples of 4 can be expressed as the difference of two squares. I'll leave that as a separate exercise for you, the mathlete. :)

Answer 3

every odd integer can be expressed as the difference of 2 perfect squares. This leaves b and d. The only even numbers that can be the difference of perfect squares seems to be those that are multiples of 4. So that rules out d since d is a multiple of 4. that leaves b as the answer

Answer 4

B and C.

Think of it this way, for any square of an integer, the number must end in 1, 4, 5, 6, or 9. (e.g., 9x9, 8x8, 7x7, etc.). It's easy to compare all the possible combinations of differences in numbers ending in those digits. The resulting end-digits are 0, 1, 2, 3, 4, 5, and 8.

That leaves numbers ending in 6 or 7 as possible candidate answers. As the above answer list hints, it doesn't matter what the rest of the number is.

Answer 5

I don't have your full answer, but the only candidates are b and d. Every odd integer is the difference of two squares.
(n+1)^2-n^2==2n+1 which can be any odd integer, for the correct value of n.

Answer 6

wow um i guess i'll go with the first guy...
um is it supposed to mean something that the numbers are close to the digits of pi... esp the first one...
3.14159265358...