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

let, 4p+5q=a be equation 1

5p+6q=a + 13 be equation 2

Multiplying eq1 by 5 and eq2 by 4 and then subtracting eq2 from eq1 we get :

q = a - 52

By trial and error :

Case 1 : q = 2, then a = 54

Case 2 : q = 3, then a = 55 and so on.

Let us substitute the values of case 1 in eq1

we get 4p + 10 = 54

p = 11

Therefore p,q = 11,2 which are prime numbers.

Therefore the value of a = 54