Diophantine Equation - Basic

Number Theory Level 1

Let $p$ and $q$ be primes that satisfy $5p + 3q = 19$ .

Find $p \times q$ .

The answer is 6.

1 solution

Venkata Karthik Bandaru
Nov 18, 2015

It is evident that one of $p, q$ must be even, because if both are even or both are odd, $5p + 3q$ will be even, which isn't true.

Now, since $p,q$ are primes and $2$ is the only even prime, we have 2 cases :

Case-I : If $q = 2$ , $p = \dfrac{19-3(q)}{5} = \dfrac{13}{5}$ , which can't be true as $p$ is a prime too.

Case-II : If $p = 2$ , $q = \dfrac{19-5(p)}{3} = 3$ .

Therefore, $(p, q) = (2, 3)$ is the only possible solution, and hence $pq = 3 \times 2 = \boxed{6}$ .

Moderator note:

Great observation that one of the primes must be 2. The problem could be improved by choosing a larger total, since otherwise we just need to check $p = 1, 2, 3, 4$ .

Diophantine Equation - Basic

The answer is 6.

1 solution

Moderator note:

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