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Question

For what least value of n, where n is a natural number :

A.5^n is divisble by 3?

B.24^n is divisible by 8?

C.15^n is divisible by 5?

D.19^n is divisible by 9?

E.18^n is divisible by 6?

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Solution

Answer :

Here n is a natural number ,

A ) 5n is divisible by 3

At any value of n , 5n is not divisible by 3 , As we can check
As we know 5 is a prime number .
So,
at n = 3 we get

5× 5 ×53 , not divisible by 3
So,
We can say that at any value of n 5n is not divisible by 3 as we now 5 is not divisible by 3 .


B ) 24n is divisible by 8

Here we can say that at any value of n 24n is divisible by 8

As we know 24 = 8 × 3

So as we can as much 24 as we want but that is always divisible by 8 , As we can check that

At n = 3

24 ×24 × 248 = 1728

So we can say that at any value of n 24n is divisible by 8
But the least value of n = 1 as 24 is also divisible by 8 ,

AS : 248 = 3
SO,
least value of n = 1

C ) 15n is divisible by 5

Here we can say that at any value of n , 15n is divisible by 5

As we know 15 = 5 × 3

So as we can as much 15 as we want but that is always divisible by 5 , As we can check that

At n = 3

15 ×15 ×155 = 675
But the least value of n = 1 as 15 is also divisible by 5 ,

AS : 155 = 3
SO,
least value of n = 1
So we can say that at any value of n, 15n is divisible by 5 .

D ) 19n is divisible by 9

At any value of n , 19n is not divisible by 9 , As we can check

As we know 19 is a prime number
So,
at n = 3 we get

19× 19 ×199 , not divisible by 9
So,
We can say that at any value of n 19n is not divisible by 9 as we now 19 is not divisible by 9.

E ) 18n is divisible by 6

Here we can say that at any value of n , 18n is divisible by 6

As we know 18 = 6 × 3

So as we can as much 18 as we want but that is always divisible by 3 , As we can check that

At n = 3

18 ×18 ×186 = 972

So we can say that at any value of n, 18n is divisible by 6 .

But the least value of n = 1 as 18 is also divisible by 6 ,

AS : 186 = 3
SO,
least value of n = 1

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