Counting Factors

'Counting Factors' printed from https://nrich.maths.org/

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Counting Factors printable sheet

Charlie wants to know how many factors 360 has.
How would you work it out?

Click below to see what Alison did.

Alison divided 360 by each number in turn to see if it was a factor, and wrote down the factor pairs:

(1, 360)
(2, 180
(3, 120)
(4, 90)
(5, 72)
(6, 60)
(8, 45)
(9, 40)
(10, 36)
(12, 30)
(15, 24)
(18, 20)

"I can stop there, because the next factor would be 20 and I've already got that. So there are 24 factors."

Charlie thought about it in a different way. Click below to see what he did.

Charlie started by working out the prime factorisation of 360.

$$\begin{align} 360 &= 2 \times 180 \\ &= 2 \times 2 \times 90 \\ &= 2 \times 2 \times 2 \times 45 \\ &= 2 \times 2 \times 2 \times 3 \times 15 \\ &= 2 \times 2 \times 2 \times 3 \times 3 \times 5 \end{align}$$

So $360 = 2^3 \times 3^2 \times 5$.

Then he made a table to list the 24 possible combinations of the prime factors.

$2^0$

$3^0$

$5^0$

$5^1$

$3^1$

$5^0$

$5^1$

$3^2$

$5^0$

$5^1$

$2^1$

$3^0$

$5^0$

$5^1$

$3^1$

$5^0$

$5^1$

$3^2$

$5^0$

$5^1$

$2^2$

$3^0$

$5^0$

$5^1$

$3^1$

$5^0$

$5^1$

$3^2$

$5^0$

$5^1$

$2^3$

$3^0$

$5^0$

$5^1$

$3^1$

$5^0$

$5^1$

$3^2$

$5^0$

$5^1$

So the top branch gives us $2^0 \times 3^0 \times 5^0 =1$
the second branch gives us $2^0 \times 3^0 \times 5^1 =5$
the third branch gives us $2^0 \times 3^1 \times 5^0 =3$
the fourth branch gives us $2^0 \times 3^1 \times 5^1 =15$...
... and the eleventh branch gives us $2^1 \times 3^2 \times 5^0 = 18$

When she saw Charlie's method, Alison said "There must be lots of numbers which have exactly 24 factors!"

Charlie and Alison think all of these numbers have exactly 24 factors.
Can you use Charlie's method to explain why?

$25725 = 5^2 \times 3^1 \times 7^3$
$217503 = 11^1 \times 13^3 \times 3^2$
$312500 = 5^7 \times 2^2$
$690625 = 17^1 \times 13^1 \times 5^5$
$94143178827 = 3^{23}$

Here are some questions to consider:

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?

Extension:

What is the smallest number with exactly 100 factors?

Which number less than 1000 has the most factors?

You may be interested in the other problems in our Hidden Treasures Feature.