Science:Math Exam Resources/Courses/MATH312/December 2010/Question 04

MATH312 December 2010

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Question 04
Suppose a positive integer n has 77 positive divisors (1 and n count among them). How many of these divisors can be prime? Explain your answer.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Start off by using the Fundamental Theorem of Arithmetic to write $\displaystyle n=\prod _{i=1}^{n}p_{i}^{\alpha _{i}}$ How can we write the number of divisors of n in this notation?

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Start off by using the Fundamental Theorem of Arithmetic to write $\displaystyle n=\prod _{i=1}^{n}p_{i}^{\alpha _{i}}$ Then the number of divisors of n is $\displaystyle d=\prod _{i=1}^{n}(\alpha _{i}+1)$ To see this, notice that each prime factor has $\displaystyle \alpha _{i}+1$ choices of occurring in the divisor of n (either you do not select the divisor or you select the divisor and some power of it). As d is given to be 77, and 77 has only the prime factors 7 and 11, we see that there can only be two nontrivial terms in the expansion of d. Thus only two of these divisors can be prime.

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