The sum of two numbers is 168. If their HCF is 14, how many such pairs of numbers are possible?

To find the number of pairs, let the two numbers be 14a and 14b, where 'a' and 'b' are co-prime integers. This is because the Highest Common Factor (HCF) is given as 14. According to the problem, the sum of these numbers is 168, so 14a + 14b = 168. Dividing the entire equation by 14, we get a + b = 12. We must now find pairs of co-prime positive integers (a, b) such that their sum is 12. The possible pairs for (a, b) are (1, 11), (2, 10), (3, 9), (4, 8), (5, 7), and (6, 6). Among these, only (1, 11) and (5, 7) are co-prime, meaning their only common factor is 1. The other pairs share common factors like 2 or 3, which would change the HCF of the original numbers. Thus, there are exactly 2 such pairs.