Divide 3

Number Theory Level 1

111 is divisible by 3.

222 is divisible by 3.

333 is divisible by 3.

444 is divisible by 3.

555 is divisible by 3.

Let $M$ be a positive integer with all the same digits but none of its digits are 3.

True or false:

If $M$ has $N$ digits, where $N$ is divisible by 3, then $M$ must be divisible by 3.

False True

1 solution

Aaryan Maheshwari
Feb 4, 2018

$M$ is divisible by $3$ if the sum of its digits is a multiple of $3$ .

As per question, let it have $3k$ digits.

So, the sum of the digits of $M=\space 3k\space \times\space N=3(Nk),$ which is obviously divisible by $3.$