A Variation on the Algorithm for GCD and LCM

1982 ◽  
Vol 30 (3) ◽  
pp. 46
Author(s):  
Verna M. Adams
Keyword(s):  
Prime Numbers ◽  
Prime Divisors ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Prime Factors

An algorithm sometimes presented for finding the least common multiple (LCM) of two numbers uses tbe technique of simultaneously finding the prime factors of the numbers. This technique is shown in figure 1. Both numbers are checked for divisibility by 2, then by 3, by 5, and so on. If the divisor does not divide one of the numbers, the number is written on the next line as shown in steps 4 and 5. This process continues until all numbers to the left and on the bottom are prime numbers, or it can be continued, as shown in figure 1, until the numbers across the bottom are all ones. The least common multiple is the product of all of the prime divisors. Thus, LCM (80, 72) = 24 · 32 · 5.

A simplified presentation for finding the LCM and the GCF

1974 ◽  
Vol 21 (5) ◽  
pp. 415-416
Author(s):  
Laurence Sherzer
Keyword(s):  
Common Factor ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Prime Factors ◽  
Positive Integers

Given the prime factors of two positive integers, the least common multiple (LCM) of these two numbers is the product of the union of these prime factors, and the greatest common factor (GCF) is the product of the intersection of these prime factors. If we could just state this fact to our students and be understood, our job of teaching them to find the LCM or the GCF of two numbers would be greatly simplified. Unfortunately, as in most teaching, simple verbal statements do not suffice.

On Nonsingular Power LCM Matrices

2006 ◽  
Vol 13 (04) ◽  
pp. 689-704 ◽  
Author(s):  
Shaofang Hong ◽  
K. P. Shum ◽  
Qi Sun
Keyword(s):  
Singular Number ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Closed Set ◽  
Prime Factors ◽  
The Matrix ◽  
Lcm Matrix ◽  
Positive Integers ◽  
Power Lcm Matrix

Let e ≥ 1 be an integer and S={x1,…,xn} a set of n distinct positive integers. The matrix ([xi, xj]e) having the power [xi, xj]e of the least common multiple of xi and xj as its (i, j)-entry is called the power least common multiple (LCM) matrix defined on S. The set S is called gcd-closed if (xi,xj) ∈ S for 1≤ i, j≤ n. Hong in 2004 showed that if the set S is gcd-closed such that every element of S has at most two distinct prime factors, then the power LCM matrix on S is nonsingular. In this paper, we use Hong's method developed in his previous papers to consider the next case. We prove that if every element of an arbitrary gcd-closed set S is of the form pqr, or p2qr, or p3qr, where p, q and r are distinct primes, then except for the case e=1 and 270, 520 ∈ S, the power LCM matrix on S is nonsingular. We also show that if S is a gcd-closed set satisfying xi< 180 for all 1≤ i≤ n, then the power LCM matrix on S is nonsingular. This proves that 180 is the least primitive singular number. For the lcm-closed case, we establish similar results.

scholarly journals A variant of the Hardy-Ramanujan theorem

2022 ◽  
Vol Volume 44 - Special... ◽  
Author(s):  
M. Ram Murty ◽  
V Kumar Murty
Keyword(s):  
Natural Number ◽  
Positive Constant ◽  
Independent Interest ◽  
Prime Divisors ◽  
Normal Order ◽  
Common Multiple ◽  
Least Common Multiple

For each natural number $n$, we define $\omega^*(n)$ to be the number of primes $p$ such that $p-1$ divides $n$. We show that in contrast to the Hardy-Ramanujan theorem which asserts that the number $\omega(n)$ of prime divisors of $n$ has a normal order $\log\log n$, the function $\omega^*(n)$ does not have a normal order. We conjecture that for some positive constant $C$, $$\sum_{n\leq x} \omega^*(n)^2 \sim Cx(\log x). $$ Another conjecture related to this function emerges, which seems to be of independent interest. More precisely, we conjecture that for some constant $C>0$, as $x\to \infty$, $$\sum_{[p-1,q-1]\leq x} {1 \over [p-1, q-1]} \sim C \log x, $$ where the summation is over primes $p,q\leq x$ such that the least common multiple $[p-1,q-1]$ is less than or equal to $x$.

Sharing Teaching Ideas

1988 ◽  
Vol 81 (8) ◽  
pp. 648-652
Author(s):  
Paul A. Klein
Keyword(s):  
Common Factor ◽  
Common Denominator ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Prime Factors ◽  
Instructional Sequences ◽  
Teaching Ideas

Few arithmetic skills have eluded as many students as those used in reducing fractions or in obtaining the least common denominator of two fractions. In the more popular instructional sequences the general concepts of greatest common factor (GCF) and least common multiple (LCM) are developed first Usually, these concepts involve the decomposition of the numbers into prime factors as shown in example 1.

THERE DOES NOT EXIST AN ODD SINGULAR NUMBER OF THE FORM plqr

2008 ◽  
Vol 01 (01) ◽  
pp. 77-83 ◽  
Author(s):  
Shaofang Hong ◽  
Shaofang Wang
Keyword(s):  
Positive Integer ◽  
Singular Number ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Closed Set ◽  
Prime Factors ◽  
Singular Numbers

Motivated by his solution to the Bourque-Ligh conjecture on the nonsingularity of the least common multiple matrix defined on the gcd-closed set, Hong introduced the concept of primitive singular number. Meanwhile Hong proved that there does not exist a singular number with no more than two distinct prime factors. Hong and Shum as well Sun showed in 2006 that there are even primitive singular numbers of the form plqr, where p, q, r are distinct primes and l is a positive integer. In this paper, we show that there does not exist an odd singular number of the form plqr. This improves a result obtained by Hong and Shum as well Sun and also confirms partially a conjecture of Hong.

Geometric Figures Make the LCM Obvious

1987 ◽  
Vol 34 (7) ◽  
pp. 17-18
Author(s):  
Flo McEnery Edwards
Keyword(s):  
Common Factor ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Geometric Figures ◽  
Prime Factors

The greatest common factor (GCF) is easily found by comparing the prime factors of a set of numbers and choosing the prime factors they have in common. The least common multiple (LCM) is often more difficult for students to find. The prime factors of the LCM are not necessarily common to a set of numbers, but their product i a common multiple of those numbers. According to Troutman and Lichtenberg (1982, 173), “the least multiple that is common to some set of numbers will have all the factors of those numbers and no other factors.” How, then, is it possible to find these prime factors just by inspection? One way is to write the prime factors of a set of numbers in a list and use geometric figures to find the prime factors of the LCM.

scholarly journals PEMBELAJARAN MATEMATIKA REALISTIK DI SEKOLAH DASAR PADA MATERI FPB DAN KPK DENGAN MODEL PENYAJIAN PAKET MAKANAN

2017 ◽  
Vol 1 (2) ◽  
pp. 113
Author(s):  
Fachrurazi Fachrurazi
Keyword(s):  
Mathematics Education ◽  
Prime Numbers ◽  
Realistic Mathematics Education ◽  
Greatest Common Divisor ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Prime Factorization ◽  
Realistic Mathematics ◽  
Learning Mathematics ◽  
Food Package

So far, the concept of the Greatest Common Divisor (GCD) and the Least Common Multiple (LCM) of two numbers are still being taught procedurally whether it is seeking multiples or by understanding prime numbers and prime factorization. This way often makes the students difficulties in determining the GCD and LCM. To overcome such things, the authors will offer a solution in learning mathematics material GCD and LCM in elementary school taught with Approach Realistic Mathematics Education (RME) food package presentation model. Mathematical learning with this approach is believed to make students learn as if they find themselves GCD and LCM concepts learned and will make learning more meaningful so that ultimately will improve students' understanding on GCD and LCM concepts.

scholarly journals On the least common multiple of random q-integers

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Carlo Sanna
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
Probabilistic Model ◽  
Expected Value ◽  
Random Set ◽  
Common Multiple ◽  
Least Common Multiple

AbstractFor every positive integer n and for every $$\alpha \in [0, 1]$$ α ∈ [ 0 , 1 ] , let $${\mathcal {B}}(n, \alpha )$$ B ( n , α ) denote the probabilistic model in which a random set $${\mathcal {A}} \subseteq \{1, \ldots , n\}$$ A ⊆ { 1 , … , n } is constructed by picking independently each element of $$\{1, \ldots , n\}$$ { 1 , … , n } with probability $$\alpha $$ α . Cilleruelo, Rué, Šarka, and Zumalacárregui proved an almost sure asymptotic formula for the logarithm of the least common multiple of the elements of $${\mathcal {A}}$$ A .Let q be an indeterminate and let $$[k]_q := 1 + q + q^2 + \cdots + q^{k-1} \in {\mathbb {Z}}[q]$$ [ k ] q : = 1 + q + q 2 + ⋯ + q k - 1 ∈ Z [ q ] be the q-analog of the positive integer k. We determine the expected value and the variance of $$X := \deg {\text {lcm}}\!\big ([{\mathcal {A}}]_q\big )$$ X : = deg lcm ( [ A ] q ) , where $$[{\mathcal {A}}]_q := \big \{[k]_q : k \in {\mathcal {A}}\big \}$$ [ A ] q : = { [ k ] q : k ∈ A } . Then we prove an almost sure asymptotic formula for X, which is a q-analog of the result of Cilleruelo et al.

scholarly journals Metaplectic flavor symmetries from magnetized tori

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Yahya Almumin ◽  
Mu-Chun Chen ◽  
Víctor Knapp-Pérez ◽  
Saúl Ramos-Sánchez ◽  
Michael Ratz ◽  
...  
Keyword(s):  
Zero Mode ◽  
Metaplectic Groups ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Yukawa Couplings ◽  
Flavor Symmetries ◽  
Euler’S Theorem ◽  
Modular Transformations ◽  
Analytic Expressions ◽  
Geometric Picture

Abstract We revisit the flavor symmetries arising from compactifications on tori with magnetic background fluxes. Using Euler’s Theorem, we derive closed form analytic expressions for the Yukawa couplings that are valid for arbitrary flux parameters. We discuss the modular transformations for even and odd units of magnetic flux, M, and show that they give rise to finite metaplectic groups the order of which is determined by the least common multiple of the number of zero-mode flavors involved. Unlike in models in which modular flavor symmetries are postulated, in this approach they derive from an underlying torus. This allows us to retain control over parameters, such as those governing the kinetic terms, that are free in the bottom-up approach, thus leading to an increased predictivity. In addition, the geometric picture allows us to understand the relative suppression of Yukawa couplings from their localization properties in the compact space. We also comment on the role supersymmetry plays in these constructions, and outline a path towards non-supersymmetric models with modular flavor symmetries.

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