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Thirty Six Exactly

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

The number $12 = 2^2 \times 3$

has 6 factors. What is the smallest natural number with exactly 36 factors?

This is how Graeme Gilpin from Newcastle Royal Grammar School arrived at the answer:

I worked out what was the lowest number that had 12 factors and came to 60. I then tried numbers over 200 that were multiples of 2, 3 and 5, e.g. 210. I realised after about an hour that no number under 1000 had exactly 36 factors. I then started again with the 2, 3 and 5 sequence until I got to a number that worked, which is 1260. All in all it took me about one and a half hours to work out the answer.

The factors are : 1260, 1, 630, 2, 420, 3, 315, 4, 252, 5, 210, 6, 180, 7, 140, 9, 126, 10, 105, 12, 90, 14, 84, 15, 70, 18, 63, 20, 60, 21, 45, 28, 42, 30, 36, 35.

Ken Nisbet from Madras College, St Andrew's wrote:
"36 Exactly was done by the whole class 3XP over several days and developed with ideas from the class. The crucial 'formula' was discovered by Dorothy Winn. Needless to say it was thereafter referred to as the 'Winning Formula'! I have sent what I considered the best four write-ups although each student in the class produced a write up for this question."

A systematic approach, which works in general, depends on prime factorisation as class 3XP discovered. Thank you to Claire Kruithof, Catherine Aitken, and Joe Neilson for your splendid write-ups. Well done 3XP!

Here is Dorothy's write-up:
"We were asked to find the smallest positive integer with exactly 36 factors. I started to look for numbers using trial and error. I took smaller numbers with lots of factors, like 36, and multiplied them by other numbers with many factors to see if I could get a larger number with even more factors. I didn't get very far with that though and I began to make a table which had all the numbers up to 40, their factors, prime factors and number of factors. I colour coded it to make it easier to see numbers which had the same number of factors.

Number Prime factor Factors No. of factors
1 1 1 1
2 2 1, 2 2
3 3 1, 3 2
4 2 2 1, 2, 4 3
5 5 1, 5 2
6 2, 3 1,2,3,6 4
Etc.

We found that there is a connection between the number of factors and the prime factorisation. I found that if you add one to each of the indices and multiply these numbers together you get the total number of factors."

[Editor's note: For example, $(2,1)$

Indices No. of factors
(1) 2
(2) 3
(1,1) 4
(3) 4
(1,3) 8
(1,1,1) 8


(1,2,5) 36
(2,2,3) 36
(17,1) etc. 36

You can do it backwards by finding the smallest numbers that multiply together to get 36, subtracting one from each of them and using them as indices. The smallest numbers that multiply together to get 36 are (2, 2, 3, 3). If you subtract one from each of them you get (1, 1, 2, 2). If you use these as indices for primes the smallest number you can get is $7^1 \times 5^1 \times 3^2 \times 2^2$ which comes to 1260. I have put the numbers in that order (7, 5, 3, 2) because the answer will be less if take lower powers of the higher numbers and higher powers of the lower numbers, squaring 2 and 3 and not 5 and 7.

You can also get 36 factors with (4,3,3). This gives$2^3 \times 3^2 \times 5^2 = 1800$ a number greater than 1260, or (2, 2, 9) which gives you$2^8 \times 3 \times 5 = 3840$ , much greater than 1260.

[ Editors note: others wrote out all the index patterns for 36 factors, and also all the factors of 1260 written as products of the prime numbers 2, 3, 5 and 7.]