The lengths of all sides of a right-angled triangle are expressed in integers, while the length of one

The lengths of all sides of a right-angled triangle are expressed in integers, while the length of one of the legs is expressed by a prime number greater than 3. What remainders when divided by 12 can a number expressing the length of the other leg give?

By the Pythagorean theorem, the ratio of the lengths of the sides of a right-angled triangle is:
a² + b² = c²;
Let b be prime, then
b² = c² – a² = (c – a) (c + a);
since b is prime, b² can only be divisible by b, b² and 1;
The factors are not equal, so the smaller one is 1 and the larger one is b²;
Therefore b² = c + a;
Then c – a = 1;
c + a = b²;
c = (b² + 1) / 2;
a = (b² – 1) / 2;
a = ((b – 1) (b + 1)) / 2;
(b – 1) (b + 1) is divisible by 3, since b is not divisible, and one of three neighboring numbers is divisible by 3;
this product is divisible by 8, since b is odd, two adjacent to it are even, and one must be divisible by 4;
Therefore, b² – 1 is divisible by 24, and – is divisible by 12, the remainder is 0;
Answer: The number expressing the length of the other leg is divisible by 12 without a remainder;