Relationship between H.C.F. and L.C.M.

We will learn the relationship between H.C.F. and L.C.M. of two numbers.

First we need to find the highest common factor (H.C.F.) of 15 and 18 which is 3.

Then we need to find the lowest common multiple (L.C.M.) of 15 and 18 which is 90.

H.C.F. × L.C.M. = 3 × 90 = 270

Also the product of numbers = 15 × 18 = 270

Therefore, product of H.C.F. and L.C.M. of 15 and 18 = product of 15 and 18.

Again, let us consider the two numbers 16 and 24

Prime factors of 16 and 24 are:

16 = 2 × 2 × 2 × 2

24 = 2 × 2 × 2 × 3

L.C.M. of 16 and 24 is 48;

H.C.F. of 16 and 24 is 8;

L.C.M. × H.C.F. = 48 × 8 = 384

Product of numbers = 16 × 24 = 384

So, from the above explanations we conclude that the product of highest common factor (H.C.F.) and lowest common multiple (L.C.M.) of two numbers is equal to the product of two numbers

or, H.C.F. × L.C.M. = First number × Second number

or, L.C.M. = $\frac{\textrm{First Number} \times \textrm{Second Number}}{\textrm{H.C.F.}}$

or, L.C.M. × H.C.F. = Product of two given numbers

or, L.C.M. = $\frac{\textrm{Product of Two Given Numbers}}{\textrm{H.C.F.}}$

or, H.C.F. = $\frac{\textrm{Product of Two Given Numbers}}{\textrm{L.C.M.}}$

Solved examples on the relationship between H.C.F. and L.C.M.:

1. Find the L.C.M. of 1683 and 1584.

Solution:

First we find highest common factor of 1683 and 1584

$Relationship between H.C.F. and L.C.M.$

Therefore, highest common factor of 1683 and 1584 = 99

Lowest common multiple of 1683 and 1584 = First number × Second number/ H.C.F.

= $\frac{1584 × 1683}{99}$

= 26928

2. Highest common factor and lowest common multiple of two numbers are 18 and 1782 respectively. One number is 162, find the other.

Solution:

We know, H.C.F. × L.C.M. = First number × Second number then we get,

18 × 1782 = 162 × Second number

$\frac{18 × 1782}{162}$ = Second number

Therefore, the second number = 198

3. The HCF of two numbers is 3 and their LCM is 54. If one of the numbers is 27, find the other number.

Solution:

HCF × LCM = Product of two numbers

3 × 54 = 27 × second number

Second number = $\frac{3 × 54}{27}$

Second number = 6

$Relation Between HCF and LCM$

4. The highest common factor and the lowest common multiple of two numbers are 825 and 25 respectively. If one of the two numbers is 275, find the other number.

Solution:

We know, H.C.F. × L.C.M. = First number × Second number then we get,

825 × 25 = 275 × Second number

$\frac{825 × 25}{275}$ = Second number

Therefore, the second number = 75

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