Problem 7. How many solutions in integers does the equation

$6 y^{2}+3 x y+x+2 y-72=0$ have?
Answer: 4.

## Solution:

Factorize:

$(3 y+1)(2 y+x)=72$.

The first factor must give a remainder of 1 when divided by 3. The number 72 has only 4 divisors that give such a remainder: $-8,-2,1,4$. They provide 4 solutions.

Problem 7a. How many solutions in integers does the equation

$6 y^{2}+3 x y+x+2 y+180=0$ have?

Answer: 6.

## Solution:

Factorize:

$(3 y+1)(2 y+x)=180$.

The first factor must give a remainder of 1 when divided by 3. The number 180 has only 6 divisors that give such a remainder: $-20,-5,-2,1,4,10$. They provide 6 solutions.