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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.

    :lol: :lol:

  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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