Answer:

x=59,-59

y=12

Step-by-step explanation:

615 + x2 = 2y

First we know x ≥ 0, so 615 ≤ 2y, and so y > 9 since 29 = 512.

In order for the two sides to be equal, the last digit of each side must be equal. So let’s start by analyzing the last digit of 615 + x2.

Last digit of

x – 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

x2 – 0, 1, 4, 9, 6, 5, 6, 9, 4, 1

From the last calculation we can add 5 to get the last digit of 615 + x2:

Last digit of

615 + x2 – 5, 6, 9, 4, 1, 0, 1, 4, 9, 6

The powers of 2 have a pattern. The last digit cycles every 4 terms as 2, 4, 8, 6 starting with y = 1, 2, 3, 4, …

In order that the last digits are equal, we want 615 + x2 to have an even last digit. Hence we must have a last digit of 4 or 6.

The last digit of 4 or 6 only occurs when y is 2, 4, …, or when y is even.

So we have established that y is even, or equal to 2N for some integer N. How does this help us? It is actually the entire key to this problem!

We can now write:

615 + x2 = 2y

615 + x2 = 22n

615 = 22n – x2

Now we can do a neat algebra trick. The right hand side is a difference of two square numbers, so we can use the formula A2 – b2 = (a + b)(a – b). So we get:

615 = 22n – x2

615 = (2n + x)(2n – x)

We now consider the factors of 615. Since 615 = 3 × 5 × 41, there are only a few possibilities:

615 = 1 × 615

615 = 3 × 205

615 = 5 × 123

615 = 15 × 41

We can further limit the possibilities because if the two factors are (2n + x) and (2n – x), then we have

sum of factors

(2n + x) + (2n – x) = 2(2n) = 2n+1

So we need the sum of the factors to be a power of 2.

615 = 1 × 615, sum(factors) = 616

615 = 3 × 205, sum(factors) = 208

615 = 5 × 123, sum(factors) = 128

615 = 15 × 41, sum(factors) = 56

The only sum that is a power of 2 is 128. So we have 128 = 27 = 2n+1, which means n = 6 and so y = 2n = 12.

How to solve for x? An easy way is to use the difference of factors:

difference of factors

(2n + x) – (2n – x) = 2x

or

(2n – x) – (2n + x) = -2x

So we have 123 – 5 = 118 = 2x, MEANING x = 59; or we have 123 – 5 = 118 = -2x, meaning x = -59.

And we’ve proven the only integer solutions are (x = 59, y = 12) and (x = -59, y = 12). We can also verify:

615 + 59^2 = 4096 = 2^12

615 + (-59)^2 = 4096 = 2^12