Latin Numbers

Can you create a Latin Square from multiples of a six digit number?

Repeaters

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

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Stage: 3 Short Challenge Level:
$3$

The units digit of the product is $3$; therefore the remainder when that number is divided by $5$ is also $3$.

This problem is taken from the UKMT Mathematical Challenges.