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Problem 236

AMC 10/12, early questions

Number theory Difficulty 3.9 Find the answer South African Mathematics Olympiad · South Africa

How many positive factors of $128$ are not factors of $120$ ?

Your answer

Official solution

$128$ has factors that are all powers of $2$ , i.e. $2^0, 2^1, \ldots, 2^7$ .

The largest power of $2$ that divides $120$ is $2^3$ .

The powers of $2$ dividing $128$ and exceeding $8$ are $2^4, 2^5, 2^6$ and $2^7$ .

So, there are $4$ positive factors of $128$ that are not factors of $120$ .

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