Magic trick

Number Theory Level 3

Leo, the magician asks a member of the audience to write down at most seven distinct prime numbers ( $>2$ ) without showing them to Leo. Then the chosen member multiply each number with itself, and tells the amount of the perfect squares to Leo. From that in a few secundum (without using calculator) Leo finds out how many numbers did the member write down.

For example if Sarah was the chosen member, and she writes down the numbers $3, 13, 23, 29,$ then she tells the number $3^2+13^2+23^2+29^2=1548$ .

Imagine that now you are the magician and you do the same trick (as Leo did) to your friend, Bob. Bob tells you the number $550632557$

How many numbers did Bob write down?

6 2 7 1 4 3 5 Can't be determined

1 solution

Joe Mansley
May 26, 2018

The square of an odd prime must be 1 mod 6, so we just need to take Bob's number mod 6, and we have the number of primes.

Not quite true - one of the primes could be $3$ , which causes some ambiguity. If you use mod $4$ in a similar way, though, you can do it in this case; either the number written down is congruent to $1$ mod $4$ , so there must be either $1$ or $5$ primes in the list; but square numbers can't end in $7$ , so the answer must be $5$ .

However, I think it works to use both mod $6$ and mod $4$ together to resolve ambiguous cases.

Chris Lewis - 1 year, 6 months ago

Magic trick

1 solution

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