Divisors
The list below shows the first ten numbers together with their divisors (factors):
1 = 1
2 = 1, 2
3 = 1, 3
4 = 1, 2, 4
5 = 1, 5
6 = 1, 2, 3, 6
7 = 1, 7
8 = 1, 2, 4, 8
9 = 1, 3, 9
10 = 1, 2, 5, 10
What is the smallest number with exactly twelve divisors?
What is the smallest number with exactly fourteen divisors?
Can you find any patterns?
How did you go about solving this problem?
A square number always has an odd number of divisors.
156=
1-156
2-78
3-52
4-39
6-26
12-13
Great work Jake and Mikaela. When I saw your answer Jake, I thought, you’ve got it but within two minutes, Kelsie came along and showed me another solution which really surprised me with how small her number was.
I was going to get her to post it but when I woke up early this morning, for some reason this problem got stuck in my head and so I thought I’d better type out what I came up with otherwise I’ll never get back to sleep.
Kelsie’s answer = 60
* 1,2,3,4,5,6,10,12,15,20,30,60,
Having looked at this, I thought the key must be 3,4,5 and so started looking at numbers ending in 0 and which these were divisors of.
70 (no) – 80 (no) – 90 (no) – 100 (no) 110 (no) 120 (yes)
It made sense to me straight away as it was double 60 and so it would have exactly the same divsors plus perhaps a few more.
1,2,3,4,5,6,8,10,12,15,30,40,60,120 which gives us 14.
Is this the smallest number with 14? I’m not certain.
If I followed the pattern, then I thought that by doubling 120, it should give me 16 divisors.
Does it?
It also got me thinking, what is the smallest number that has 1,2,3,4,5,6,7,8,9,10 as divisors? Looks a tad hard for me, especially at this time in the morning 😉
The number I found was:
3268800
I found this by multiplying all of the numbers together. Im not sure if this is the lowest but it is a possibilty.
After looking at my answer closer I thought why times all of them together when I can just multiply the multiples of them all .So,
intead of multiplying 8,2, and 4 I can just multiply it by 8 because 8 is a multiple of both 2 and 4. The same thing for 9 and 3, and 6,2 and 3.
Still not sure if this is the lowest but its much lower than my previous answer.