Think of a number from 1 to 9 and multiply it by 15873. Multiply
your answer by 7 and the product's 6 numerals are all the same you started
with.
Think of a number from 1 to 9 and multiply it by 37. Multiply your answer by 3 and the product's 3 numerals are all the same you started with. 

Explanation below
When you first multiply your number by 15873 and then by 7, you are actually multiplying your single digit number by 15873 x 7 = 111111. So, if your digit is 4, then 4 x 111111 = 444444.
In other words, 4 x 15873 x 7 = 4 x (15873 x 7) = 4 x (111111) = 444444.
If your number is 9, then 9 x 15873 x 7 = 9 x (15873 x 7) = 9x (111111) = 999999
When you first multiply your number by 37 and then by 3, you are actually multiplying your single digit number by 37 x 3 = 111. So, if your digit is 4, then 4 x 111 = 444.
In other words, 4 x 37 x 3 = 4 x (37 x 3) = 4 x (111) = 444.
If your number is 9, then 9 x 37 x 3 = 9 x (37x 3) = 9x (111) = 999.
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