CEMC Fermat

Number theory Difficulty 2.4 Junior Find the answer

How many of the positive divisors of 128 are perfect squares larger than 1?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Since $128=2^{7}$ , its positive divisors are $2^{0}=1, 2^{1}=2, 2^{2}=4, 2^{3}=8, 2^{4}=16, 2^{5}=32, 2^{6}=64, 2^{7}=128$ . Of these, the integers $1,4,16,64$ are perfect squares, which means that 128 has three positive divisors that are perfect squares larger than 1.

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