Problem 5

A deck of

n>1

cards is given. A positive integer is written on each card. The deck has the property that the arithmetic mean of the numbers on each pair of cards is also the geometric mean of the numbers on some collection of one or more cards. For which

n

does it follow that the numbers on the cards are all equal?

Step 7 of 8: The geometric mean uses only p-free cards

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In plain words

A prime-free product cannot contain a prime-divisible factor.

x^m=\prod_{r=1}^{m}a_{i_r},\qquad p\nmid x\Longrightarrow p\nmid a_{i_r}

Detailed analysis

By the hypothesis, $x$ is the geometric mean of $m\ge1$ cards. Thus $x^m=\prod_{r=1}^m a_{i_r}$ . Since $p\nmid x$ , no factor is divisible by $p$ , so every factor is at most $a_k$ .