Two natural numbers are in the ratio 6 : 11. If their L.C.M. is 462, then…

For two natural numbers in ratio a : b, where a and b are co-prime (share no common factor), if H is their H.C.F., the numbers can be written as aH and bH. Since a and b are co-prime, the L.C.M. of these two numbers is simply a × b × H.

1. Let the two given numbers be 6x and 11x, where x is their H.C.F. (6 and 11 are co-prime, since their only common factor is 1).

2. Because 6 and 11 share no common factor, the L.C.M. of 6x and 11x is 6 × 11 × x = 66x.

3. The problem states the L.C.M. is 462, so 66x = 462.

4. Solving, x = 462 ÷ 66 = 7. Hence the H.C.F. of the two numbers is 7.

As a check, the two numbers are 6 × 7 = 42 and 11 × 7 = 77. Factorising, 42 = 2 × 3 × 7 and 77 = 7 × 11, so their L.C.M. is 2 × 3 × 7 × 11 = 462 (matching the given value) and their H.C.F. is 7 (the only common prime factor) — confirming the answer.