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 = 2^{3} × 3

The factors of 18 = 2 × 3 × 3 = 3^{2} × 2

The highest power of 2 = 2^{3}

The highest power of 3 = 3^{2}

Therefore, the lowest common multiple of 18 and 24 is 2^{3} × 3^{2} = 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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