Question

a number namely n is composed of 9 digits.All of which are same n is definitely divisible by 111,27,11,9,37

Answer 1

Answer:

n is definitely divisible by 111, 9, 37

Step-by-step explanation:

As the number n has 9 digits all the same, it is automatically divisible by 111, 9 and 37.  Let's see why...

The number n is a multiple of k = 111111111.

The sum of the digits in k is 9 => k is divisible by 9 => n is divisible by 9.

k = 111111111 = 111000000 + 111000 + 111 = 111 × 1001001 => k is divisible by 111 => n is divisible by 111.

k is divisible by 111 = 3 × 37  =>  k is divisible by 37  =>  n is divisible by 37

The remaining two numbers --- 27 and 11 --- are not necessarily factors of n.

k / 9 = 12345679.  The sum of the digits here is 37, which is not divisible by 3  =>  k/9 is not divisible by 3  =>  k is not divisible by 27  =>  n is not guaranteed to be divisible by 27.  n is only divisible by 27 if the common digit is 3, 6 or 9, so introduce the extra factor of 3.

Let the common digit be d.

Then n = dk = d(111111110 + 1) = 11×10101010×d + d

so the remainder when n is divided by 11 is d.

Therefore no matter what d is, n is never divisible by 11.