Question

In how many ways can 80 be written as the difference of squares of two natural numbers?

• A. 2
• B. 3
• C. 4
• D. 5

Answer 1

The correct option is B

3

To write a number as a difference of squares of 2 natural numbers, we need to find the 2 factor products of the number, such that both the factors are either even or odd.

This is because if we are writing a number N as a difference of squares, then -
$N=a^2-b^2⟹N=(a+b)(a-b)$. If (a+b, a-b) is a pair of one odd and one even number, 'a' and 'b' can't be natural numbers.
Hence, both the factors need to be either even or odd.
In the case of 80, the possible cases are $80=2\times 40;=4\times 20;=8\times 10.$
Thus, there are only three instances where both are even. Hence, the answer is option (b).