Question

If p and q are prime numbers such that 4p+5q=a and 5p+6q=a+13, where a is a positive integer, then what is a

Answer 1

First of all, we can solve the two equations simultaneously.

$5p+6q=a+13 \\4p+5q=a$

If we subtract equation 2 from equation 1, we get: $p + q = 13$.

The only two prime numbers which satisfy this equation are 2 and 11. Now we need to know which is p and which is q.

Assume that p =2 and q = 11. Therefore, if we substitute in the 2nd equation:

$4p+5q=4(2)+5(11)=a=63$.

Now we need to check from the 1st equation: $5p+6q=5(2)+6(11)=76=a+13$. Which is correct.

Therefore, a=63.