Perfect Square Ratio

Number Theory Level 2

What is the smallest positive integer $N$ for which $\dfrac{2016}{N}$ is a perfect square ?

The answer is 14.

1 solution

Paola Ramírez
Aug 18, 2016

As the smallest $N$ wants to be found, it is needed to look for the largest square which can be obtained by prime decomposition of $2016=2^5\times3^2\times7$ . The largest square obtained is $2^4\times3^2$ so the smallest $N$ is $14$

Moderator note:

Great! For clarity, can you explain how you obtained the value $2^4\times3^2$ ?

The prime factorization of 2016 can be stated as 7 x 3 x 3 x 2 x 2 x 2 x 2 x 2. For a number to be a perfect square, it must be the product of two equal numbers; and so in order to divide 2016 down to a number which is a perfect square, we'll need to divide it down to a number which has two of each of its prime factors. This occurs when we divide out a 2 and a 7, so 14 is the number we need to divide by.

Ryan Thomas - 4 years, 9 months ago

Perfect Square Ratio

The answer is 14.

1 solution

Moderator note:

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