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A certain number of people are sitting in a circle, all facing The Mathemagician who is standing in the center of the circle. Each is given a calculator, paper, and a pencil. The Mathemagician declares:

(Note: the letters N and A through E below are substituted for their actual values when they are spoken below.)

I want all of you to write down any number of length N on a piece of paper.

Now write the same number again next to your number to make a longer number.

Now divide your longer number by A, write the result on a scrap of paper and hand it to the person on your right.

Now take the number on the scrap you just received, divide that number by B, write the result on a scrap of paper and hand it to the person on your right.

Now take the number on the scrap you just received, divide that number by C, write the result on a scrap of paper and hand it to the person on your right.

Now take the number on the scrap you just received, divide that number by D, write the result on a scrap of paper and hand it to the person on your right.

Now take the number on the scrap you just received, divide that number by E, write the result on a scrap of paper and hand it to the person on your right.

Now raise your hand if you have just been handed your original number.

Find the smallest value of N such that if A, B, C, D, and E are certain prime numbers, then at least one person will raise their hand. Your answer must specify the values of N, A, B, C, D, E, as well as the number of people in the circle to satisfy the requirements.

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One possibility:

There are 5 people. $N=9.$ $\{A, B, C, D, E\} = \{7, 11, 13, 19, 52579\}$, this being the factorisation of $10^9 + 1$.

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    $\begingroup$ Well done. That was fast. Though I knew it would be easy for those who knew what to look for.,, $\endgroup$ – TTT Jun 10 '16 at 18:15

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