Track / Stage 3 / 248 of 260 #248 of 1964

Problem 248

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

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

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 $120$ .

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.