Mathematics · Algebra
GCF and LCM Calculator
Find the greatest common factor and least common multiple of two whole numbers.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Apply the Euclidean algorithm to 12 and 18: GCF = 6.
- Compute |12 × 18| ÷ 6.
- LCM = 36.
Understand GCF and LCM
One idea, three depths
Choose how deeply to explain GCF and LCM
GCF and LCM: Find the greatest common factor and least common multiple of two whole numbers.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using GCF and LCM to answer this question: find the greatest common factor and least common multiple of two whole numbers? Enter First whole number and Second whole number; the calculator shows Greatest common factor. For example: For 12 and 18, GCF = 6 and LCM = 36. The answer tells you Greatest common factor.
Age 15Explain it to a 15-year-oldConnect it to the formula
The GCF captures every shared prime factor, while the LCM contains enough factors to be divisible by both numbers. The rule is LCM(a,b) = |ab| ÷ GCF(a,b). Its input values are First whole number, Second whole number, and the main result is Greatest common factor. For example: For 12 and 18, GCF = 6 and LCM = 36.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated gcf and lcm relation over the valid integer domain stated below. The implemented relation is LCM(a,b) = |ab| ÷ GCF(a,b), evaluated from First whole number, Second whole number to produce Greatest common factor. The GCF captures every shared prime factor, while the LCM contains enough factors to be divisible by both numbers. The GCF divides both inputs; the LCM is divisible by both inputs.
Inputs and valid domain
- First whole number must be an integer, at least 1.
- Second whole number must be an integer, at least 1.
Important boundary: The GCF divides both inputs; the LCM is divisible by both inputs.
The formula
LCM(a,b) = |ab| ÷ GCF(a,b)
How the calculator works through it
It substitutes First whole number, Second whole number into the formula and exposes every numerical step above. The main output is Greatest common factor, accompanied by Least common multiple.
Read the result correctly
The Greatest common factor is the direct answer to “find the greatest common factor and least common multiple of two whole numbers.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
For 12 and 18, GCF = 6 and LCM = 36.
Where this model stops being reliable
The GCF divides both inputs; the LCM is divisible by both inputs.
Learn it by changing one value
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Dictionary terms behind this calculator
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.
These foundations help you understand why GCF and LCM works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
GCF and LCM uses LCM(a,b) = |ab| ÷ GCF(a,b). You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.
Review this foundation about 4 min
Strong support
- Functions and input-output rules
A function viewpoint helps you see how changing an input changes the GCF and LCM result.
Review this foundation about 5 min
Optional enrichment
- Powers and exponents
Powers are not required for every GCF and LCM calculation, but they make related algebraic forms and code easier to read.
Review this foundation about 4 min
Mathematics → algorithm → program
Implement this calculation in code
These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.
Algorithm
- Use absolute input values.
- Repeatedly replace (a,b) by (b,a mod b).
- When b becomes zero, a is the greatest common factor.
Python
from math import gcd
def greatest_common_factor(a: int, b: int) -> int:
return gcd(a, b)
assert greatest_common_factor(12, 18) == 6
C
#include <assert.h>
#include <stdint.h>
int64_t greatest_common_factor(int64_t a,int64_t b){if(a<0)a=-a;if(b<0)b=-b;while(b){int64_t r=a%b;a=b;b=r;}return a;}
int main(void){assert(greatest_common_factor(12,18)==6);}
C++
#include <cassert>
#include <numeric>
long long greatest_common_factor(long long a,long long b){return std::gcd(a,b);}
int main(){assert(greatest_common_factor(12,18)==6);}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · integer arguments in rdi, rsi and rdx
; int64_t greatest_common_factor(int64_t a, int64_t b)
global greatest_common_factor
section .text
greatest_common_factor:
mov rax, rdi
test rax, rax
jns .a_ok
neg rax
.a_ok:
mov rcx, rsi
test rcx, rcx
jns .b_ok
neg rcx
.b_ok:
test rcx, rcx
jz .done
.loop:
xor rdx, rdx
div rcx
mov rax, rcx
mov rcx, rdx
test rcx, rcx
jnz .loop
.done:
ret
MATLAB
function result = greatest_common_factor(a, b)
result = gcd(abs(round(a)), abs(round(b)));
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a_Integer, b_Integer] := GCD[a, b];
Continue in mathematical software
The downloaded file includes your current inputs and first calculated result. It is created locally.
Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.
Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations
Standards, reading and academic references
Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.
Algebra and Trigonometry 2e
Read the related free OpenStax mathematics chaptersCite this book
- APA 7
- Abramson, J. (2021). Algebra and trigonometry 2e. OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites
- MLA 9
- Abramson, Jay. Algebra and Trigonometry 2e. OpenStax, 2021, https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.
- Chicago author-date
- Abramson, Jay. 2021. Algebra and Trigonometry 2e. Houston, TX: OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.
OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.
Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS
These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.
APA 7
MW SysArc. (2026, July 21). GCF and LCM Calculator. MW SysArc Tools. https://math.mwsysarc.com/algebra/gcf-lcm-calculator
MLA 9
MW SysArc. “GCF and LCM Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/algebra/gcf-lcm-calculator. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “GCF and LCM Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/algebra/gcf-lcm-calculator.
Harvard
MW SysArc (2026) ‘GCF and LCM Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/algebra/gcf-lcm-calculator (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_gcf_lcm_2026,
author = {{MW SysArc}},
title = {GCF and LCM Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/algebra/gcf-lcm-calculator},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - GCF and LCM Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/algebra/gcf-lcm-calculator
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the GCF and LCM do?
Find the greatest common factor and least common multiple of two whole numbers.
How does the GCF and LCM work?
The calculator applies LCM(a,b) = |ab| ÷ GCF(a,b). The GCF captures every shared prime factor, while the LCM contains enough factors to be divisible by both numbers.
What can I learn from the GCF and LCM?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed . Calculations tested .