## 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.

April 5th, 2011 at 12:17 am

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.

April 5th, 2011 at 3:45 am

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

April 5th, 2011 at 9:17 am

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.

April 5th, 2011 at 9:26 am

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

April 5th, 2011 at 2:20 pm

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?

April 5th, 2011 at 2:36 pm

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.

April 5th, 2011 at 8:12 pm

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

April 9th, 2011 at 12:28 am

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.

April 9th, 2011 at 12:29 am

;-;

April 9th, 2011 at 5:46 am

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.

April 11th, 2011 at 10:45 am

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.

April 12th, 2011 at 5:42 am

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

April 14th, 2011 at 12:22 pm

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.

April 22nd, 2011 at 9:45 am

1 2 3 4 6 9 12 18 27 36 54 108

the totle number is 12

April 22nd, 2011 at 9:46 am

1 2 3 4 6 9 12 18 27 36 54 108

the total number is 12