The product of two positive integers is 2028 and their HCF is 13. How many…

Concept

When two positive integers have HCF h, write them as ha and hb, where gcd(a, b) = 1.

After removing the common factor h from both numbers, the product condition becomes h × h × ab = P. Only coprime factor pairs of P/(h × h) are admissible.

Application

1. Let the integers be 13a and 13b with gcd(a, b) = 1.

2. (13a)(13b) = 2028, so 169ab = 2028 and ab = 12.

3. The positive factor pairs of 12 are (1, 12), (2, 6), and (3, 4). Their gcd values are 1, 2, and 1 respectively, so the coprime pairs are (1, 12) and (3, 4).

4. Multiplying each cofactor by 13 gives (13, 156) and (39, 52).

Cross-check

The products are 13 × 156 = 2028 and 39 × 52 = 2028; each pair has HCF 13. Reversing a pair does not create a new unordered pair.

Therefore, the number of unordered pairs is 2.