The H.C.F. of two numbers is 23 and the other two factors of their L.C.M. are 13 and 14. Find the larger number.

Given data

• H.C.F. = 23
• The co-prime parts contributing to L.C.M. are 13 and 14.

Concept / Approach

• If HCF = h and the co-prime parts are p and q, the numbers can be taken as h·p and h·q (since gcd(p, q) = 1).

Step-by-step calculation

Numbers: 23×13 = 299 and 23×14 = 322L.C.M. = 23×13×14 and H.C.F. = 23, as required.

Verification

gcd(299, 322) = 23 (since 299 = 23×13 and 322 = 23×14).

Common pitfalls

• Multiplying 13 and 14 again into a single number, which is not the intent.

Final Answer

Larger number = 322.