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?

Answer 1

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

$\frac{5\times 5\times 5}{3}$ , 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 $\times$ 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

$\frac{24\times 24\times 24}{8}$ = 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 : $\frac{24}{8}$ = 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 $\times$ 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

$\frac{15\times 15\times 15}{5}$ = 675
But the least value of n = 1 as 15 is also divisible by 5 ,

AS : $\frac{15}{5}$ = 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

$\frac{19\times 19\times 19}{9}$ , 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 $\times$ 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

$\frac{18\times 18\times 18}{6}$ = 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 : $\frac{18}{6}$ = 3
SO,
least value of n = 1