Question

Can two numbers have 16 as their HCF and 380 as their LCM? Give reason.

Answer 1

We know that

HCF - Highest Common Factor

For two numbers, their HCF is the largest number which divides both.

Again, LCM - Least Common Multiple

For two numbers, their LCM is the smallest number which is a multiple of both.

Now, HCF divides both numbers. Further, both numbers perfectly divide LCM.

So, HCF must perfectly divide LCM.

In other words, for LCM÷HCF, remainder must be zero.

Here, LCM = 380
And, HCF = 16

Clearly, $\frac{380}{16}$ = 23.75 i.e. 23 $\frac{3}{4}$. Clearly the remainder is not a zero.

Thus, there does not exist two numbers whose HCF is 16 and LCM is 380.

To find: Can two numbers have 16 as their HCF and 380 as their LCM?

On dividing 380 by 16 we get 23 as the quotient and 12 as the remainder

Since LCM is not exactly divisible by the HCF, two number cannot have 16 as their HCF and 380 as their LCM