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B Factoring a number and prime numbers?

  1. Apr 20, 2017 at 4:04 AM #1
    A number can be factored into a product of its component factors
    A number can be factored into a product of its prime .
    But, What exactly is a prime number ?
    Prime numbers are numbers greater than 1 that are evenly divisible only by themselves and 1

    Is it a number that can only be evenly divided by that number itself and one ?As in like it cannot be factored further?
     
    Last edited: Apr 20, 2017 at 4:19 AM
  2. jcsd
  3. Apr 20, 2017 at 4:10 AM #2
    Yes, if one could factor primes further then they would be divisible by another number and hence wouldn't be primes. In fact beyond that every number can be uniquely factored into its components, which is one of the fundamental theorems of number theory.
     
  4. Apr 20, 2017 at 4:13 AM #3
    Ok , Thanks . so its a bit like this ?

    Prime Numbers

    The first few prime numbers are 2, 3, 5, 7, 11, 13 .
    A prime number is a positive integer which has no factors other than 1 and itself. 1 itself, by definition, is not a prime number.
    Prime numbers cant be divided any further and thus can be thought of as the atoms of numbers.
    Any number which is not prime can be written as the product of prime numbers, we simply keep dividing it into more parts until all factors are prime

    84 = 2 x 2 x 3 x 7

    ?
     
  5. Apr 20, 2017 at 4:22 AM #4
    Yes and as I mentioned above every number can be factored into its prime factors in one way only. This is also the reason why 1 is not condsidered a prime number.
     
  6. Apr 20, 2017 at 4:34 AM #5
    Thanks a lot

    I am not sure where to from here , can i move to algebra and practice some problems ?

    Does algebra involve prime factorization ?
     
    Last edited: Apr 20, 2017 at 4:48 AM
  7. Apr 20, 2017 at 4:46 AM #6
    No, I do not think so. I believe that prime factorization is part of pre-algebra and number theory. The only connection I can think of is factorizing polynomials, which may involve thinking about the factors of a number.
    I am not sure which type of questions you are looking for. If you are looking for questions, which involve simply factorizing numbers I would recommend a website called " Brilliliant.org". They offer problems and information on nearly all branches of maths and offer those kinds of practice problems.
     
  8. Apr 20, 2017 at 4:49 AM #7
  9. Apr 20, 2017 at 4:51 AM #8
    Sometimes i don't even know what to search for .

    Thanks a lot
     
  10. Apr 20, 2017 at 12:20 PM #9

    fresh_42

    Staff: Mentor

    "Component factors" isn't a well-defined term. The correct term would be "irreducible factors" to emphasize that those cannot be split further. The others are called reducible.
    This is a theorem, which holds in some kind of number sets and which says: irreducible numbers are prime numbers. It is not automatically true, e.g. the real numbers don't have any primes.
    A number is called prime, if whenever it divides a product, it has to divide (at least) one of the factors: ##p\,\vert \,a\cdot b \Longrightarrow p\,\vert \,a \, \vee \, p\,\vert \,b##.
    This means that primes are always irreducible but not necessarily all irreducible elements are prime.
    Yes, because it can be shown that irreducible and prime are the same in the integers. The fact that unities like ##\pm 1## are ruled out is deliberately made. The purpose to do so is to formulate theorems in a reasonable way, since one can always add arbitrary many ones or evenly many minus ones to a factorization without changing the result. If unities (elements with a multiplicative inverse) were allowed, it would be harder to get to the core meaning of prime numbers or unnecessarily complicated to formulate results: "Let ##p## be a prime, but not a unity, then ..." would have been to added almost everywhere.
     
    Last edited: Apr 20, 2017 at 1:58 PM
  11. Apr 21, 2017 at 2:52 AM #10
    I am a bit confused ,i was trying to learn polynomial factorization . but before that , i thought i would refresh my arithmetic factorization skills . these things are confusing me a lot .

    Especially terms like , factors , prime factors ,factorize , factorization


    number factors.
    algebraic factors

    not sure which term belongs to which or what term should be used where ...


    If you write a polynomial as the product of two or more polynomials, you have factored the polynomial.

    equations.png

    Here is an example

    algebraic_factors.png

    how do i factor a polynomial ?

    factoring_quadratic_equation.png

    The steps involved ...

    polynomial_factorization.png

    anyway what my main aim is to learn about polynomial factorization itself , i hope that's what it is actually called ... :)
     
  12. Apr 21, 2017 at 10:05 AM #11

    Mark44

    Staff: Mentor

    factors -- expressions that can be multiplied to produce some given expression. For example, 4 and 9 are factors of 36, because 4 * 9 = 36. Also 2x and 5x are factors of 10x2, because (2x)(5x) = 10x2.
    prime factors - expressions that can't be further broken up into simpler factors. For example, the prime factors of 36 are 2, 2, 3, and 3, since 2 * 2 * 3 * 3 = 36. 4 and 9 are factors of 36, but they aren't prime factors. Both 2 and 3 are prime numbers (or primes). A number (integer) is prime if its only factors are 1 and the number itself.
    factorize (or factor) - (verb) to reduce an expression into a product (multiplication) of subexpressions. In the US we typically say "factor an expression" rather than "factorize an expression." This shorthand usage might be confusing to some, because "factor" can be used as both a noun (e.g., "7 is a factor of 49") and as a verb (e.g., "factor 15 into a product of prime numbers.").
    In your previous post you factored the trinomial ##6x^2 + 19x + 16## into a product of two binomial factors ##(2x + 3)(3x + 5)##.
    factorization - (noun) the action of reducing an expression into a product of its factors.
     
  13. Apr 21, 2017 at 11:06 AM #12
    Thanks a lot for explaining a lot of things Mark44 , it nice to most of the important points in one thread :)
     
  14. Apr 22, 2017 at 3:10 PM #13
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