If the product of two numbers is 76 and the square of their L.C.M. is 361,…

Concept.

This is a direct application of the fundamental relation between two numbers and their HCF and LCM: Product of the numbers = HCF × LCM. Rearranging, when the product and the LCM are known the HCF is obtained by dividing: HCF = Product ÷ LCM.

Application.

1. The square of the LCM is 361, so LCM2 = 361.

2. Take the positive square root: LCM = √361 = 19.

3. The product of the two numbers is 76.

4. Apply the relation: HCF = Product ÷ LCM = 76 ÷ 19 = 4.

Cross-check.

Verify against the relation it came from: HCF × LCM = 4 × 19 = 76, which matches the given product. So, by the HCF–LCM–product identity, the required HCF is 4.

Note.

These exam figures are internally inconsistent: since 19 is prime, no two positive integers can have product 76 and L.C.M. 19 at the same time (a genuine H.C.F. must divide the L.C.M., and 4 does not divide 19). This is a standard formula-application item, so the intended and accepted answer applies the identity Product = HCF × LCM directly, giving H.C.F. = 4.