Multiples of n

Number Theory Level 3

For positive integer, $n$ , consider the following $10$ numbers: $n, 2n, 3n, 4n, 5n, 6n, 7n, 8n, 9n, 10n$

At most how many of these can be made perfect squares by picking the right $n$ ?

5 4 3 All of them 1 2

1 solution

Geoff Pilling
Oct 18, 2018

In order for two positive integers to be able to be made into a perfect square by multiplying them both by the same positive integer, the parities of the exponents of the prime factors needs to match. And, the largest group for which this is true is $1, 4,$ and $9$ .

You can make these $3$ squares by choosing $n$ to be a square number. Then $n$ , $4n$ and $9n$ will also be perfect squares.

To be precise, the problem should ask " At most how many of these can be perfect squares", because as it stands, 1 and 2 are also correct options.

Jon Haussmann - 2 years, 7 months ago

Good point... Done

Geoff Pilling - 2 years, 7 months ago

Multiples of n

1 solution

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