If a and b are integers then the equation 8a + 15b = 45 has

The equation 8a + 15b = 45 is a linear Diophantine equation. A Diophantine equation is an equation where only integer solutions are allowed.

To find the solutions, we can use the Euclidean algorithm:

1. First, find the greatest common divisor (gcd) of 8 and 15. The gcd of 8 and 15 is 1.

2. Since 45 is divisible by the gcd, the equation has integer solutions.

3. To find a particular solution, we can use the extended Euclidean algorithm. This algorithm not only finds the gcd of two numbers, but also finds coefficients x and y such that 8x + 15y = gcd(8, 15).

4. Once we have a particular solution (a0, b0), all other solutions can be found by using the formula a = a0 + 15t and b = b0 - 8t, where t is any integer.

So, the equation 8a + 15b = 45 has infinite integer solutions.