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# Proper Factors

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Age 16 to 18

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

This problem requires pupils to think about prime number factorisation in different ways. There is no A-level content required to answer this question, but it does require careful and logical thought.

Pupils may find it easier to consider the total number of factors first, before moving onto proper factors.

Start with the first part of part **1**, i.e. ask students the question "Show that $3^2\times 5^3$ has exactly $10$ proper factors". Ask students how they might approach this. Some ideas might be:

- Calculate the factors of $1125$ (having first worked out that $3^2\times 5^3=1125$)
- Work on some easier cases first, such as $3$, $3 \times 5$, $3^2 \times 5$, and see if this provides some insights into the problem
- Can the form $3^2\times 5^3$ be used to help answer the question? Can we use this form to work out what the prime factors are?

After considering this question, pupils can then find other numbers of the form $3^m\times 5^n$ which also have $10$ proper factors.

Before asking students to think about the question in part **2**, they might like to try and find the smallest number which has exactly 10 proper factors.

Here is a list of number theory problems.