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Problem 213

Number theory Difficulty 1.6 Multiple choice CEMC Gauss (Grade 7) · Canada · 2012

The number 503 is a prime number. How many positive integers are factors of 2012?

Pick one

Official solution

Both 1 and 2012 are obvious positive integer factors of 2012.

Since 2012 is an even number and $2012\div2=1006$ , then both 2 and 1006 are factors of 2012.

Since 1006 is also even then 2012 is divisible by 4.

Since $2012\div4=503$ , then both 4 and 503 are factors of 2012.

We are given that 503 is a prime number; thus there are no additional factors of 503 and hence there are no additional factors of 2012.

The factors of 2012 are 1 and 2012, 2 and 1006, 4 and 503.

Therefore, there are 6 positive integers that are factors of 2012.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.