What is the lowest multiple of 49 that is composed of the same digits (e.g. 444, 7777).?

Question

What is the lowest multiple of 49 that is composed of the same digits (e.g. 444, 7777). Please explain how to get this without using a calculator.

Answer 1

If the digits are not all 1's or all 7's, then they are not the lowest multiple, because you could divide by the digit value (2, 3, 4, 5, 6, 8 or 9) to get 111 . . . as a smaller multiple. This would work because none of these digits divides into 49.

If the digits are all 7's, then dividing them by 7, a string of 1's of the same length must be a multiple of 7. Take a string of 1's, and divide by 7 until there is a zero remainder. It turns out that this happens after six 1's. So a string of six 7's is a multiple of 49.

A string of forty-two 1's is a multiple of 49, but getting that is a very hard sum involving modular exponentiation and Euler's Theorem.

Answer 2

I think that it's 777,777. That's 15,873 x 49. I'm not sure, though, and don't ask me to explain how I got it, I cheated and used a calculator. :)