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 a can be?

Answer 1

4p + 5q = a   (1)

5p + 6q =a + 13 ------(2)

(1) x 5 ==> 20p + 24q = 4a + 52     --------(3)

(2) x 4 ==>20p + 25q = 5a             ---------(4)

    (4) - (3) ==> q= a -52

therefore (1) becomes ap + (a - 52)x5 = a

4p + 5a - 260 = a

4p + 4a = 260

p +a = 65

a = 65 -p

therefore 'a' is positive integer less than 65