Math # The LCM and HCF of two numbers are 60 and 3 respectively, if the first number is 12 what will the second number be?

The LCM of two numbers is 60. One of the numbers is 12. What is the hcf?

We know that LCM is the union of Prime factors of the given numbers in which common prime factors taken once including all the factors.

Normally when we are given the numbers and we have to find their LCM we find prime factors of the numbers and take the union of Prime factors.

This problem is a bit tricky as we are given only one number and the LCM.

In these types of problems we apply reverse engineering or go backwards.

We find prime factors of LCM and the given numbers and find the possible values of the unknown number by comparing them.

12 = 2 x 2 x 3

LCM= 60=2 x 2 x 3 x 5

We cn see only factor 5 in LCM which is not present in the prime factors of 12, so 5 must be in the factors of unknown number with or without the other factors of LCM.

The possible values for unknown number are:

5, 2 x 5, 3 x 5, 2 x 2 x 5, 2 x 3 x 5, 2 x 2 x 3 x 5

Or

5, 10, 15, 20, 30, 60

If the numbers are 5 and 12 then their HCF would be 1

If the numbers are 10 and 12 then their HCF would be 2.

If the numbers are 15 and 12 then their HCF would be 3.

If the numbers are 20 and 12 then their HCF would be 4.

If the numbers are 30 and 12 then their HCF would be 6

If the numbers are 60 and 12 then their HCF would be 12.

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