CEMC Fermat, 2016

Number theory Difficulty 1.4 Junior Find the answer Canada

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

Pick one

Solution

Since $128 = 2^7$ , its positive divisors are $2^0=1 \qquad 2^1=2 \qquad 2^2=4 \qquad 2^3=8 \qquad 2^4=16 \qquad 2^5=32 \qquad 2^6=64 \qquad 2^7=128$ 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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