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 Write-up

Title
Magic Pairs Problem

Problem Statement
The pair of numbers 3 and 37037 is "magic." Here is how the magic works: Pick your favorite digit from among 1, 2, 3, or 9, and multiply it by 3. Multiply the result by 37037. What is the final answer?

Is the pair of numbers 13 and 8547 "magic?" pick your favorite digit from among 1, 2, 3, or 9, and multiply it by 13. Multiply the result by 8547. What is the final answer?

How many pairs of "magic" numbers exist?

Problem setup

I multiply 1, 2, 3, or 9 by 3.  Next, that product is multiplied by 33037.  What is the answer?  What would allow the numbers 3 and 37037 to be be called "magic"?

I multiply 1, 2, 3, or 9 by 13.  Next, that product is multiplied by 8547.  What is the answer?  What would allow the numbers 13 and 8547 to be called "magic"?

Plans to Solve/Investigate the Problem

I'll take each possible example and look for a pattern.  Then I will extrapolate my results in an attempt to add more "magic" pairs of numbers.

Investigation/Exploration of the Problem

1 * 3 * 37037 = 111111

2 * 3 * 37037 = 222222

3 * 3 * 37037 = 333333

9 * 3 * 37037 = 999999

Thus far, the pattern is clear.  Obviously 3 * 37037 = 111111.  What would happen is we tried a few more?

4 * 3 * 37037 = 444444

5 * 3 * 37037 = 555555

13 * 3 * 37037 = 1444443

10 * 3 * 37037 = 1111110

I would predict that in order to be "magic", the original chosen factor has be to be less than 10, whether it be even or odd.  Beyond the factor 9, the pattern does not hold true.

Extensions of the Problem