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  • ...f Arithmetic]] tells us that every [[positive integer]] has a unique prime factorization, up to changing the order of the terms. The form of a prime factorization is
    3 KB (496 words) - 21:14, 5 January 2024
  • #REDIRECT [[Prime factorization]]
    33 bytes (3 words) - 14:54, 19 June 2006
  • ...etic]] holds. More precisely an integral domain <math>R</math> is a unique factorization domain if for any nonzero element <math>r\in R</math> which is not a [[unit ...n domain. This automatically implies that many well-known rings are unique factorization domains including:
    6 KB (1,217 words) - 22:05, 23 August 2009

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  • This factorization frequently shows up on contest problems, especially those heavy on algebrai
    4 KB (682 words) - 12:13, 8 December 2024
  • If <math>n=2</math>, this creates the difference of squares factorization, <cmath>a^2-b^2=(a+b)(a-b)</cmath> This leads to the difference of cubes factorization, <cmath>a^3-b^3=(a-b)(a^2+ab+b^2)</cmath>
    3 KB (532 words) - 21:00, 13 January 2024
  • ==Prime Factorization== {{main|Prime factorization}}
    10 KB (809 words) - 15:40, 17 March 2024
  • Given the general [[prime factorization]] of <math>{n} = {p}_1^{e_1}{p}_2^{e_2} \cdots {p}_m^{e_m}</math>, one can To derive the formula, let us first define the [[prime factorization]] of <math> n </math> as <math> n =\prod_{i=1}^{m}p_i^{e_i} =p_1^{e_1}p_2^{
    5 KB (903 words) - 14:49, 27 July 2024
  • A number is divisible by <math>N</math>, where the [[prime factorization]] of <math>N</math> is <math>p_1^{e_1}p_2^{e_2}\cdots p_n^{e_n}</math>, if The prime factorization of 36 to be <math>2^2\cdot 3^2</math>. Thus we must check for divisibility
    10 KB (1,572 words) - 21:11, 22 September 2024
  • ...for every factor <math>(a_nx^n+a_{n-1}x^{n-1}+\cdots +a_0)^m</math> in the factorization of <math>Q(x)</math>, introduce the terms
    2 KB (414 words) - 01:02, 13 February 2009
  • === Using prime factorization === Once the [[prime factorization]]s of the given numbers have been found, the greatest common divisor is the
    2 KB (288 words) - 21:40, 26 January 2021
  • ...2b+1)=2(2ab+a+b)+1</cmath> and if <math>a,b>0</math> this is a non-trivial factorization. ...imes ([[permutation|permutations]] not withstanding). This unique [[prime factorization]] plays an important role in solving many kinds of [[number theory]] proble
    6 KB (1,036 words) - 17:26, 2 September 2024
  • ...f Arithmetic]] tells us that every [[positive integer]] has a unique prime factorization, up to changing the order of the terms. The form of a prime factorization is
    3 KB (496 words) - 21:14, 5 January 2024
  • ...dot p_{2}^{\alpha_{2}}\cdot\dots\cdot p_m^{\alpha_m}</math> is the [[prime factorization]] of <math>{n}</math>, then the number <math>d(n)</math> of different divis
    1 KB (274 words) - 18:50, 29 August 2023
  • === Using prime factorization === The LCM of two or more numbers can also be found using [[prime factorization]]. In order to do this, factor all of the numbers involved. For each [[pr
    2 KB (383 words) - 09:49, 4 September 2022
  • ==Use in factorization== <math>i</math> is often very helpful in factorization. For example, consider the difference of squares: <math>(a+b)(a-b)=a^2-b^2<
    2 KB (321 words) - 14:57, 5 September 2008
  • ...th>, and using the [[Sum and difference of powers | difference of powers]] factorization yields <cmath>S(r-1) = a_1(r-1)(1 + r + r^2 + \cdots + r^{n-1}) = a_1(r^n-1
    4 KB (649 words) - 20:09, 19 July 2024
  • ...video link for detailed explanation of the proof and the concept of unique factorization: https://youtu.be/jfDbnz-Bp_g
    3 KB (453 words) - 10:13, 9 June 2023
  • ...of <math>6n\pm 1</math> factors so only if one of variables is 0 will the factorization be trivial (contain only 1 and itself).
    2 KB (308 words) - 01:27, 1 May 2024
  • <math>\mathbb{Z}</math>: the [[integer]]s (a [[unique factorization domain]]). ...h>a \in \mathbb{Z}</math> and odd <math>n= \prod p_i^{\nu_i}</math> (prime factorization of <math>n</math>) is defined as: <math>\left( \frac{a}{n} \right) = \prod
    8 KB (1,401 words) - 12:11, 17 June 2008
  • ...th>p</math> satisfying <math>\frac{2n}3<p\le n</math> appears in the prime factorization of <math>\binom{2n}{n}</math> is <math>\left\lfloor\frac{2n}{p}\right\rfloo
    2 KB (317 words) - 07:44, 18 October 2024
  • Indeed, this factorization yields two solutions to the congruence: <math>x \equiv 5 \pmod{21}</math>,
    14 KB (2,317 words) - 18:01, 29 October 2021
  • ...11y^{13}.</math> The minimum possible value of <math>x</math> has a prime factorization <math>a^cb^d.</math> What is <math>a + b + c + d</math>?
    13 KB (1,987 words) - 17:53, 10 December 2022
  • ...square-free, then each prime dividing m only has a power of 1 in the prime factorization, so given (4) that m|<math>x\cdot (r-1)^2</math>, m has at most one of each
    5 KB (883 words) - 00:05, 2 June 2024

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