How many pairs of positive integers p, q exist such that the HCF of p, q is 37 and the sum of p and q is 1961?

Correct option is B

Given:

HCF(p, q) = 37

p + q = 1961

Need: Number of positive ordered pairs (p, q) satisfying the above conditions

Solution:

Let p = 37x and q = 37y, with HCF(x, y) = 1

Given p + q = 1961:

37(x + y) = 1961

x + y = 1961 ÷ 37 = 53

Since 53 is prime, every positive pair (x, y) with x + y = 53 is automatically coprime.

Possible values of x are 1, 2, 3, …, 52

→ Total pairs = 52

Thus, the number of positive integer pairs (p, q) satisfying the conditions is 52.