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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