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

This topic discusses about the relationship between H.C.F and L.C.M in case of two numbers. The relationship between H.C.F and L.C. M of two number states that:

H.C.F × L.C.M = First Number × Second Number

Hence from this relationship we can find the following:

\[\textrm{H.C.F} = \frac{\textrm{First Number} \times \textrm{Second Number}}{\textrm{L.C.M.}}\]

\[\textrm{L.C.M.} = \frac{\textrm{First Number} \times \textrm{Second Number}}{\textrm{H.C.F.}}\]

\[\textrm{First Number} = \frac{\textrm{H.C.F.} \times \textrm{L.C.M.}}{\textrm{Second Number}}\]

\[\textrm{Second Number} = \frac{\textrm{H.C.F.} \times \textrm{L.C.M.}}{\textrm{First Number}}\]

To Justify this Relation here is an Example:

1. Suppose the two given numbers are 24 and 16. Justify the relationship between H.C.F and L.C.M

Solution:

The H. C.F of 24 and 18 by finding factors is:

The factors of 24 = 2 × 2 × 2 × 3 = 2^3×3

The factors of 18 = 2 × 3 × 3 = 3^2×2

The common prime factors of 24 and 18 = 2 and 3 

The lowest power of 2 is 2

The lowest power of 3 is 3

Therefore, the Highest common factor (H.C.F) = 2 × 3 = 6

The L.C. M of 24 and 18 by prime factorization method:

The factors of 24 = 2 × 2 × 2 × 3 = 23 × 3

The factors of 18 = 2 × 3 × 3 = 32 × 2

The highest power of 2 = 23

The highest power of 3 = 32

Therefore, the lowest common multiple of 18 and 24 is 23 × 32 = 72

Now the product of H.C.F and L.C.M is 6 × 72 = 432

The product of 18 and 24 is 432

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

432 = 432 (verified)

Here are few more examples to illustrate relationship between H.C.F and L.C.M


2. The H.C.F of two numbers is 18 and their product is 94734. Find their L.C.M

Solution:

Given,

The highest common factor (H.C.F) of two numbers = 18

The product of two numbers = 94734

According to the relationship, we know,

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

Or, H.C.F × L.C.M = product of the numbers

⟹ 18 × L.C.M = 94734

⟹ L.C.M = \(\frac{94734}{18}\)

⟹ L.C.M = 5263

Therefore, the lowest common multiple (L.C.M) of the two numbers = 5263

Explanation:

The H.C.F of two numbers is given and their product is given. By applying the relation we need to find out the L.C.M


3. The L.C.M and H.C.F of two numbers are 144 and 36 respectively. If one number is 54, then find the other number.

Solution:

Given,

The highest common factor (H.C.F) of two numbers = 36

The lowest common multiple (L.C.M) of two numbers = 144

One of the two numbers is 54

According to the relationship, we know,

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

⟹ 36 × 144 = 54 × Second number

⟹ 5184 = 54 × Second number

⟹ Second Number = \(\frac{5184}{54}\)

⟹ Second Number = 96

Therefore, the other number is 96

Explanation:

The H.C.F and L.C.M of two numbers is given and one of the numbers is given. By applying the relationship we can find out the other/ second number.

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