Greatest Common Factor Calculator

GCF calculator

Find the greatest common factor

Enter two or more integers. The largest factor that divides every number with no remainder will appear first, followed by the method.

Input

Numbers to compare

Separate values with commas, spaces, or new lines.

Result

Greatest common factor

The answer stays prominent; the proof follows below.

GCF(24, 36)

12 divides all 2 entered integers exactly.

Method

See why the answer works

Prime factorization

Keep the factors shared by every number

242³ × 3
362² × 3²

Shared prime factors: 2² × 3 = 12.

Exact division check

Every quotient is a whole number, so every remainder is zero.

|24| ÷ 12 = 2No remainder
|36| ÷ 12 = 3No remainder

Euclidean algorithm steps

Reduce one pair at a time; the previous GCF is compared with the next number.

gcd(24, 36) = 12
24 = 36 × 0 + 2436 = 24 × 1 + 1224 = 12 × 2 + 0

Equal-cut planning bench

Cut two stock lengths into the longest identical pieces

The greatest common factor turns a shared divisibility requirement into a practical maximum. When two stock lengths must be cut into equal whole pieces with no waste, their GCF is the longest permitted piece—not merely any length that divides both.

Step-by-step guide · Equal-cut planning bench

Prime exponents explain why forty-eight is the longest safe cut

Shared divisor: The GCF divides every entered integer exactly
Prime rule: Keep the lowest exponent shared by all inputs
Cut example: gcf(96,144) = 48

Decompose each stock length
Write 96 = 2⁵×3 and 144 = 2⁴×3². Multiplying the primes reconstructs each original length and checks that the decompositions are complete.
Retain only shared prime powers
Both numbers contain powers of 2 and 3. The greatest factor cannot use more than four 2s or more than one 3, because 144 lacks a fifth 2 and 96 lacks a second 3.
Multiply and interpret
Calculate 2⁴×3 = 16×3 = 48. The rolls yield five pieces altogether: two from the first roll and three from the second.

Cut two stock lengths into the longest identical pieces: The greatest common factor turns a shared divisibility requirement into a practical maximum. When two stock lengths must be cut into equal whole pieces with no waste, their GCF is the longest permitted piece—not merely any length that divides both.

Two ribbon rolls make the maximum condition visible

A display shop has ribbon rolls measuring 96 centimeters and 144 centimeters. It needs equal-length pieces, wants no remainder, and prefers the pieces to be as long as possible. Forty-eight centimeters satisfies all three constraints: 96/48 = 2 and 144/48 = 3. A 24-centimeter cut would also divide both lengths, but it would produce shorter pieces than required. A 50-centimeter cut is longer but leaves waste, so it is not a common factor.

Listing divisors works for modest numbers. The positive factors of 96 include 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, and 96. The factors of 144 include 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, and 144. Their largest shared entry is 48. The factor calculator is useful when you want to inspect either list separately rather than jump directly to the intersection.

Prime exponents explain why forty-eight is the longest safe cut

Decompose each stock length
Write 96 = 2⁵×3 and 144 = 2⁴×3². Multiplying the primes reconstructs each original length and checks that the decompositions are complete.
Retain only shared prime powers
Both numbers contain powers of 2 and 3. The greatest factor cannot use more than four 2s or more than one 3, because 144 lacks a fifth 2 and 96 lacks a second 3.
Multiply and interpret
Calculate 2⁴×3 = 16×3 = 48. The rolls yield five pieces altogether: two from the first roll and three from the second.

Common-factor work differs from multiple and fraction work

GCF maximizes an exact shared group size

The ribbon question moves downward to divisors of each fixed length. It asks how large one identical piece can be.

LCM minimizes a future shared multiple

A scheduling question moves upward through multiples. Use the least common multiple calculator when two repeating intervals must next coincide.

Fraction reduction uses a common factor but preserves a ratio

Dividing 96/144 by the GCF produces 2/3. The fraction calculator expresses that reduced relationship; it does not answer the maximum physical cut length unless the units and constraints are supplied.

Equal-cut answers depend on integer units and zero-waste assumptions

Convert both measurements to the same smallest practical unit before applying an integer GCF. Comparing 0.96 meters with 144 centimeters as the raw integers 0.96 and 144 is not the intended whole-number problem.
Account for saw kerf, trimming, knots, or seam allowance separately. Pure divisibility assumes cutting consumes no material.
The conventional GCF is nonnegative. Input signs may be ignored for divisibility magnitude, but a negative physical length remains invalid data.
Zero requires a stated convention: every nonzero integer divides zero, while gcf(0,n) is commonly |n| and gcf(0,0) is not determined by a largest positive divisor. Do not guess silently.

Important limitations

Greatest Common Factor Calculator is limited to the finite inputs, modes, and outputs stated for the Greatest common factor of integers method. Check integers, validation messages, the equation, and the stated display precision before relying on greatest common factor and integer count.

This formula passed its registered engineering tests and is published under an explicit site-owner policy authorization. No independent expert, QA, editorial, legal, or third-party review is claimed. Verify consequential results with a qualified professional.

Equal-cut planning bench

Cut two stock lengths into the longest identical pieces

Step-by-step guide · Equal-cut planning bench

Prime exponents explain why forty-eight is the longest safe cut

Shared divisor: The GCF divides every entered integer exactly
Prime rule: Keep the lowest exponent shared by all inputs
Cut example: gcf(96,144) = 48

Decompose each stock length
Write 96 = 2⁵×3 and 144 = 2⁴×3². Multiplying the primes reconstructs each original length and checks that the decompositions are complete.
Retain only shared prime powers
Both numbers contain powers of 2 and 3. The greatest factor cannot use more than four 2s or more than one 3, because 144 lacks a fifth 2 and 96 lacks a second 3.
Multiply and interpret
Calculate 2⁴×3 = 16×3 = 48. The rolls yield five pieces altogether: two from the first roll and three from the second.

Two ribbon rolls make the maximum condition visible

Prime exponents explain why forty-eight is the longest safe cut

Decompose each stock length
Write 96 = 2⁵×3 and 144 = 2⁴×3². Multiplying the primes reconstructs each original length and checks that the decompositions are complete.
Retain only shared prime powers
Both numbers contain powers of 2 and 3. The greatest factor cannot use more than four 2s or more than one 3, because 144 lacks a fifth 2 and 96 lacks a second 3.
Multiply and interpret
Calculate 2⁴×3 = 16×3 = 48. The rolls yield five pieces altogether: two from the first roll and three from the second.

Common-factor work differs from multiple and fraction work

GCF maximizes an exact shared group size

The ribbon question moves downward to divisors of each fixed length. It asks how large one identical piece can be.

LCM minimizes a future shared multiple

A scheduling question moves upward through multiples. Use the least common multiple calculator when two repeating intervals must next coincide.

Fraction reduction uses a common factor but preserves a ratio

Dividing 96/144 by the GCF produces 2/3. The fraction calculator expresses that reduced relationship; it does not answer the maximum physical cut length unless the units and constraints are supplied.

Equal-cut answers depend on integer units and zero-waste assumptions

Convert both measurements to the same smallest practical unit before applying an integer GCF. Comparing 0.96 meters with 144 centimeters as the raw integers 0.96 and 144 is not the intended whole-number problem.
Account for saw kerf, trimming, knots, or seam allowance separately. Pure divisibility assumes cutting consumes no material.
The conventional GCF is nonnegative. Input signs may be ignored for divisibility magnitude, but a negative physical length remains invalid data.
Zero requires a stated convention: every nonzero integer divides zero, while gcf(0,n) is commonly |n| and gcf(0,0) is not determined by a largest positive divisor. Do not guess silently.

Keep the factors shared by every number

Decompose each stock length

Retain only shared prime powers

Multiply and interpret

Two ribbon rolls make the maximum condition visible

Prime exponents explain why forty-eight is the longest safe cut

Common-factor work differs from multiple and fraction work

GCF maximizes an exact shared group size

LCM minimizes a future shared multiple

Fraction reduction uses a common factor but preserves a ratio

Equal-cut answers depend on integer units and zero-waste assumptions

Important limitations

Decompose each stock length

Retain only shared prime powers

Multiply and interpret

Two ribbon rolls make the maximum condition visible

Prime exponents explain why forty-eight is the longest safe cut

Common-factor work differs from multiple and fraction work

GCF maximizes an exact shared group size

LCM minimizes a future shared multiple

Fraction reduction uses a common factor but preserves a ratio

Equal-cut answers depend on integer units and zero-waste assumptions