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## Simple division

Posted by Chris on April 4, 2011 – 8:20 pm

12 can be divided exactly by 6 numbers (including 1 and 12). How many numbers can 108 be divided by? What is the general formula?

By numbers, I mean positive integers.

This one’s pretty easy, and quite possibly useful.

This post is under “MathsChallenge” and has 15 respond so far.
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### 15 Responds so far- Add one»

1. 1. Wizard of Oz Said：

108 can be divided by 11 different numbers.

For any composite number of the form prod(pi^ni) where pi are the prime factors each raised to the power ni, the number of divisors would be 1 + i*sum(ni).

That is, the number of factors times the sum of all the powers, plus 1 to cover the number 1.

So, 108 = 2^2 * 3^3, which has two factors and a total of five powers, hence 1 + 2*5 = 11.

Can’t see how this general formula would be particularly useful, except to solve problems like these.

2. 2. Chris Said：

Hi WIz. You’re getting warm. 108 can be divided by 12 numbers.

3. 3. cazayoux Said：

I see 12.

We can see it as basically 2^2 * 3^3.

For base 2, we can have a power of 0, 1, or 2.
For base 3, we can have a power of 0, 1, 2, or 3.

3*4 = 12 numbers.

4. 4. Chris Said：

Hi cazayoux. That’s close enough for the proverbial cigar

We can write any number as p1^n1 * p2^n2 * …, where pN is the Nth prime and nN its multiplicity. Then the number of divisors of the number is (n1+1)(n2+1)…

I don’t think I’d come across this straightforward fact before. It’ll be very handy if you’re asked how many divisors a number has

5. 5. John24 Said：

This seems like a great solution for smaller numbers but what if it was something much larger like 8,640,000? How would you work this one backwards?

It happens to be 2^2 * 3^3 * 4^4 * 5^5 Or does this not work since it has 4^4 with 4 not being prime? But could be 2^10 * 3^3 * 5^5 (Does this does not meet the nth prime part?)

Does this mean it has 11 * 4 * 6 = 240 factors?

The question would have to be worded in such a way so the testee understands this is a number built from a special list of prime numbers raised to a specific nth prime in order for this to be useful.

How many factors does the following special number have 2^2 * 3^3 * 5^4 * 6^5 = 524,880,000?

6. 6. Chris Said：

Hi John24. 2^2 * 3^3 * 4^4 * 5^5 is not standard prime factor form. You are right in saying that you must write it as 2^10 * 3^3 * 5^5. Then you get 11*4*6 = 264 divisors.

524,880,000 = 2^2 * 3^3 * 5^4 * 6^5 = 2^7 * 3^8 * 5^4. It has 8*9*5 = 360 divisors.

When the smallest prime factors are very large, factorising becomes very difficult – a fact that enables public key encryption to be possible.

7. 7. Chris Said：

Hi John24. I just re-read your comment. There is only one way to write a number in terms of primes (except for order) – there is no “special list”. See: http://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic

8. 8. gemma Said：

hi chris its gemma remember me
108 can be divided by 12 which aswell it can be divided by 9 cause 12,24,36,48,60,72,84,96,108.

9. 9. gemma Said：

;-;

10. 10. Chris Said：

Hi gemma. I’m not sure what your point is. Your list contains 24,48,60,72,84 and 96 – none of which divide 108.

Because 108 = 2² 3³, there are (2+1)(3+1) = 12 divisors. They are 1,2,3,4,6,9,12,18,27,36,54,108.

11. 11. Abhinav Said：

Just find the prime factors and raise them to their max powers.

eg 108=3^3 * 2^2 .

The factors will be (3+1)(2+1)=12.

12. 12. chunky Said：

12 can be divided 6 different times. 108 can be divided by 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108.

13. 13. Priyanka Said：

The answer is 12.

The basic ideology is to calculate the factors. With simple division method, we can start dividing the number by 2 until we end at 1 eventually. Now we must calculate how many times are the prime numbers are being used while dividing the given number.

For 108. We use 2 twice and 3 thrice.
thus, by the formula (p^e1)*(p^e2)
where p=prime number
e=exponential

we get (2^2)*(3^3) and so,

the number of factors are = (e1+1)*(e2+2)
= (2+1)*(3+1)
= 3*4
= 12 factors.

14. 14. rabecca Said：

1 2 3 4 6 9 12 18 27 36 54 108
the totle number is 12

15. 15. rabecca Said：

1 2 3 4 6 9 12 18 27 36 54 108
the total number is 12

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