The ratio of two numbers is 5:6 and their HCF is 12. Their LCM is

Concept: For two numbers written in a ratio a : b where a and b share no common factor other than 1 (i.e. a and b are co-prime), the numbers can be expressed as a·h and b·h, with h equal to their HCF — the shared factor between a·h and b·h is exactly h since a and b themselves share none. Their LCM is then a·b·h, because the LCM must include each ratio term once together with the shared factor h.

1. The ratio of the two numbers is 5 : 6, so let the numbers be 5x and 6x for some positive integer x.

2. Since 5 and 6 share no common factor other than 1, the HCF of 5x and 6x is x.

3. The HCF is given as 12, so x = 12, giving the two numbers 5 × 12 = 60 and 6 × 12 = 72.

4. Applying the concept above, the LCM of 5x and 6x is 5 × 6 × x = 30 × 12 = 360.

Cross-check: This can be verified independently using the identity HCF × LCM = product of the two numbers: 12 × 360 = 4320, and 60 × 72 = 4320 — the two sides match, confirming the LCM is 360.