Problem 3. Determine the prime numbers $a<b<c$ that satisfy the relation

$$
13 a+65 b+289 c=6622
$$
## Problem 3.

The numbers a, b, c cannot all be odd because the left side of the equation would be an odd number, which cannot equal 6622. Therefore, we can have either one even number or all even numbers. (2p)

If all three were even, we would have $\mathrm{a}=\mathrm{b}=\mathrm{c}=2$, which does not satisfy the equation. (1p)

Therefore, only one number is even. Since $a < b < c$, it follows that $a=2$. (1p)

We obtain: $65 b + 289 c = 6596$. Since $6596 = 17 \cdot 388$ and $289 = 17^2$, it follows that $\mathrm{b}$ must be a multiple of 17. But b is a prime number, so $b=17$. (2p). Substituting, we get $c=19$ (1p)