Question

There is a one number which is formed by writing one digit 6 times (eg.111111,555555,...etc.). such a number is always divisible by:
a-9
b-13
c-15
d-17

Answer 1

let that digit be a.
$a a a a a a = a(10^5+10^4+10^3+10^2+10+1) = 111111 * a \\ \\ Sum\ of\ alternate\ digits\ in\ 111111\ is\ 3.\ \ It\ is\ divisible\ by\ 11.\ \\ \\ sum\ of\ digits\ is\ 6.\ So,\ divisible\ by\ 3.$

aaaaaa = a * 111111 = a * 3 * 11 * 3367 = a * 3 * 11 * 13* 259
111111 is divisible by 13. So all those number are divisible by 13.

another simpler method: divisibility by 13

111111 => 11111+1*4 = 11115 => 1111 + 5*4 = 1131 => 113 + 1*4 = 117 = 9 * 13
So 111111 is divisible by 13 and so aaaaaa is also divisible.

Answer 2

555555/15=37037
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answer is fine