60^2 + x^2 = y^2

3600 = y^2 - x^2

3600 = (y + x)(y - x)

y - x = 3600 / (y + x)

Remember that x and y are positive integers.

As x gets bigger, y gets bigger, so y + x gets bigger, so y - x gets smaller.

But y is the hypotenuse so it must be bigger than x, so it y - x must be positive.

Since x and y are integers, y - x is an integers, so it must be at least 1.

Can y - x = 1? In that case:

y + x = 3600 and y = x + 1

So x + x + 1 = 3600, which means x = (3600 - 1) / 2 = 1799.5, which is not an integer, so that's impossible.

Therefore, y - x can't be less than 2.

I wrote a simple computer program to try x as every positive integer starting with 1, which yields:

11^2 + 60^2 = 61^2

25^2 + 60^2 = 65^2

32^2 + 60^2 = 68^2

45^2 + 60^2 = 75^2

63^2 + 60^2 = 87^2

80^2 + 60^2 = 100^2

91^2 + 60^2 = 109^2

144^2 + 60^2 = 156^2

175^2 + 60^2 = 185^2

221^2 + 60^2 = 229^2

297^2 + 60^2 = 303^2

448^2 + 60^2 = 452^2

899^2 + 60^2 = 901^2

Now that x is 899 and y is 901, their difference is 2, and we know from above that this difference can't go lower than 2, so we're done at this point.

Answer: 13