The H.C.F. of two numbers is 23 and the other two factors of their L.C.M. are…

When two numbers have H.C.F. h and their L.C.M. is expressed as h times two co-prime factors p and q (p and q share no common factor other than 1), the two original numbers are exactly h times p and h times q.

1. H.C.F. of the two numbers = 23; the two given co-factors of the L.C.M. are 13 and 14.

2. Check that 13 and 14 are co-prime: 13 is prime and 14 = 2 times 7, so they share no common factor other than 1 — the co-factor relationship applies directly.

3. First number = 23 times 13 = 299.

4. Second number = 23 times 14 = 322.

5. The larger of 299 and 322 is 322.

Independent check: 322 divided by 23 equals 14 exactly, confirming 23 is a common factor; since 13 and 14 are co-prime, the H.C.F. of 299 and 322 is exactly 23, matching the given condition.

Hence, the larger of the two numbers is 322.