Finding two numbers when the HCF and LCM are given

HCF and LCM

HCF or the Highest Common Factor of two numbers is the largest number that can divide both the numbers. LCM or the Least Common Multiple is the smallest number that is evenly divisible by both the numbers. LCM is also known as Least Common Divisor. Now that we know what LCM and HCF are, how do we find two numbers when only their HCF and LCM are known?

Finding two numbers when their HCF and LCM are given:

Let us take any two numbers a, b, and let us say their HCF and LCM are s and y respectively. To find out the two numbers a and b, we must first understand the relationship between the two numbers and their HCF and LCM. For any two numbers, the product of the two numbers is equal to the product of their HCF and LCM.  Let us take the example of 10 and 15. The HCF is 5 and the LCM is 30. Now, the product of the two numbers is 10✕15 = 150and this is equal to the product of the HCF and LCM which is 5 ✕ 30 = 150

Now that we have understood the relationship between the numbers and their HCF and LCM, we can find the two numbers by taking out the factors of the products of the HCF and LCM. We then eliminate the numbers which have a different HCF or LCM and any remaining pair is the number we are looking for. Note that there could be more than one pair of numbers that have the same HCF and LCM.

Taking the above example, let us find two numbers a and b, and their HCF is 5 while their LCM is 30.

Now we know that a ✕ b = 5✕30 = 150

Let us factorize the product 150.

Factors of 150 are

1 x 150,

2 x 75,

3 x 50,

5 x 30,

6 x 25, and

10 x 15

We know that the HCF of the two numbers is 5. So we can eliminate (1, 150) (2,75) (3,50) (6,25) since their HCF is not 5. we are left with (5,30) and (10,15)  and these are the two possible answers to this question. 

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