## HCF of 2472 and 1284 and a third number 'n' is 12. If their LCM is 2^{3}×3^{2}×5×103×107. Find the number 'n'.

The largest possible number which divides the given numbers exactly is called the highest common factor (HCF)

## Answer: If the HCF of 2472 and 1284 and a third number 'n' is 12 and if their LCM is 2^{3}×3^{2}×5×103×107, then, the number 'n' is 180 ( 2^{2} × 3^{2} × 5)

Let us see how to find the value of n.

## Explanation**:**

The Greatest Common Factor of any set of numbers is the largest possible number which divides the given numbers exactly without any remainder.

Given that, LCM(2472, 1284, and n) is 2^{3 }× 3^{2 }× 5 × 103 × 107

Let us express the numbers 2472, and 1284 as a product of prime numbers.

2472 = 2 × 2 × 2 × 3 × 103 (2^{3}× 3× 103 )

1284 = 2 × 2 × 3 × 107 (2^{2}× 3× 107)

HCF(2472 , 1284)= 2× 2 × 3 (2^{2}× 3)

'n' should also have one of the factors as HCF, and another factor as the missing element of other numbers from the LCM (.i.e., 3 x 5)

Therefore,

n = 2^{2} × 3 × (3 x 5)

n = 2^{2} × 3^{2} × 5

n = 4 x 9 x 5

n = 180