Counting Factors

Why do this problem?

This problem invites students to explore the power of using a prime factorisation representation of a number.

Possible approach

This problem featured in an NRICH webinar in June 2020.

Start by inviting students to work out how many factors some numbers have, perhaps including the example 360 as in the problem. (It is plausible that someone might be interested in factors of 360, so you could make the connection to angle work and regular polygons.)

Once they have had a go, share strategies for working it out; most students will probably list all the factor pairs, prompting a discussion of systematic working, and deciding when to stop. 

Students with an interest in programming may wish to consider how to write a simple program to find all the factors of a number. For very large numbers, the realisation that you only need consider potential factors less than the square root of the number speeds up a program considerably!

If no-one used a prime factorisation method similar to Charlie's, share his method from the problem and the tree-diagram or table structure he used to count the factors.

Then give students some time in pairs to discuss how we can be sure the following numbers all have exactly 24 factors:
$25725 = 5^2 \times 3^1 \times 7^3$
$217503 = 11^1 \times 13^3 \times 3^2$
$312500 = 5^7 \times 2^2$
$690625 = 17^1 \times 13^1 \times 5^5$
$94143178827 = 3^{23}$

Once students have had a chance to make sense of Charlie's method, challenge them to apply it to solve some of the questions below:

How can I find a number with exactly 14 factors?
How can I find the smallest such number?

How can I find a number with exactly 15 factors?
How can I find the smallest such number?

How can I find a number with exactly 18 factors?
How can I find the smallest such number?

Which numbers have an odd number of factors?

Possible extension

What is the smallest number with exactly 100 factors?
Which number less than 1000 has the most factors?

These, and other similar questions, could be explored with paper and pencil using prime factorisation or could be an opportunity for students to use spreadsheets or simple programming.

Students might be interested in this page from the On-line Encyclopedia of Integer Sequences which lists the smallest number with exactly n divisors.