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Multiplication Magic

Age 14 to 16 Challenge Level:

Here is something to try on your friends.

Ask a friend for a three digit number and suppose she says 314 then you immediately echo her number in your reply "Did you know that 241314 is exactly divisible by 37?"

Another friend gives 628 and you say "Amazing, 628371 is exactly divisible by 37 as well!"

When they check they find you are correct.

In general if the friend gives the number $abc$ you give $abcdef$ or $defabc$ where $a+d = b+e = c+f = x$ where $1 \leq x \leq 9$.

Why does this work?

You can even do the same trick with nine digit numbers. Your friend suggests 143 and you say, immediately "Another multiple of 37 is 143110635!"

You use exactly the same technique. Take your friend's number and think of two more three digit numbers such that the sum of the first digits, the sum of the second digits and the sum of the third digits are all the same.

Explain why the trick works for nine digit numbers.