Question

can two numbers have 24 as their HCF and 280 as their LCM ?give reason​

Answer 1

Answer:

Let the two numbers be ‘[math]a[/math]’ and ‘[math]b[/math]’. Now 16 should factor to ‘[math]a[/math]’ and ‘[math]b[/math]’. [math]a[/math] and [math]b[/math] may have many other common factors but 16 is the biggest factor. 280 is the LCM of two numbers means 280 is the multiple of [math]a[/math] and also [math]b[/math]. [math]a[/math] and [math]b[/math] may also have other common multiples but 280 is the smallest common multiple.

If 16 is factor to [math]a[/math] and [math]b[/math] and 280 is multiple of [math]a[/math] and [math]b[/math] as common logic says 16 should be factor to 280. But we know that 16 is not a factor of 280. Hence two numbers can not have 16 as their HCF and 280 as their LCM

Answer 2

Answer:

There can be two numbers having 24 as their HCF and 280 as their LCM.

Reason:

We know that the product of two numbers us equal to the product of their HCF and LCM.

Let the two numbers be a and b.

Then a x b = HCF x LCM

a x b = 24 x 280

Now, for 24 to be the HCF of a and b, both a and b must be multiple of 24.

So, let a be the smallest multiple of 24,

that is 24 itself.

So, 24 x b = 24 x 280

This gives b = 280

So, a = 24 and b = 280

But the HCF of these two numbers is 8

and not 24.

Thus, a cannot be equal to 24.

Also, if we take any bigger multiple of 24,

then aslo we will arrive at a contradiction.

Thus, there cannot exit two natural numbers having 24 as their HCF and 280 as their LCM.