GCF of 45 and 60 = 15
Quick Answer
GCF(45 and 60) = 15.
Common divisors: 1, 3, 5, 15. Related LCM: 180.
GCF Calculator
Step-by-Step: Euclidean Algorithm
The Euclidean algorithm finds the GCF by repeated division: divide the larger by the smaller, replace with the remainder, repeat until the remainder is 0. The last non-zero divisor is the GCF.
| Step | Dividend ÷ Divisor | Quotient | Remainder |
|---|---|---|---|
| 1 | 60 ÷ 45 | 1 | 15 |
| 2 | 45 ÷ 15 | 3 | 0 |
GCF(45 and 60) = 15
Common Divisors of 45 and 60
Every common divisor of 45 and 60 is a divisor of their GCF (15). The complete list of 4 common divisors:
1, 3, 5, 15
See the full divisors of 15 (which IS the GCF and contains all 4 common divisors).
Related: LCM of 45 and 60
LCM(45, 60) = 180
Derived from the identity LCM(a, b) × GCF(a, b) = a × b:
LCM = (45 × 60) ÷ GCF = 2700 ÷ 15 = 180
See the dedicated LCM of 45 and 60 page for the full common-multiples list and step-by-step derivation.
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How to Find the Greatest Common Factor
The Greatest Common Factor (GCF) of two or more numbers is the largest positive integer that divides each of the numbers without leaving a remainder.
Method 1: Euclidean Algorithm (used above)
- Divide the larger number by the smaller. Note the remainder.
- Replace the pair (larger, smaller) with (smaller, remainder).
- Repeat until the remainder is 0. The last non-zero divisor is the GCF.
Method 2: Listing Factors
- List all factors of each number
- Identify the factors that appear in all lists (common factors)
- Select the largest common factor — this is the GCF
Method 3: Prime Factorization
- Find the prime factorization of each number
- Identify which prime factors appear in all factorizations
- For each common prime factor, take the lowest exponent
- Multiply these prime factors raised to their lowest exponents
GCF = product of (common prime factors)lowest exponent
Example: GCF(12, 18) using prime factorization
- 12 = 2² × 3¹
- 18 = 2¹ × 3²
- Common primes: 2 and 3 — take the lowest exponent of each
- GCF = 2¹ × 3¹ = 6
Frequently Asked Questions
What is the GCF of 45 and 60?
GCF(45 and 60) = 15. This is the largest integer that divides each of the inputs without remainder.
How do you find the GCF of 45 and 60?
Apply the Euclidean algorithm (see the step-by-step table above). For 3 numbers, reduce pairwise: GCF(a, b, c) = GCF(GCF(a, b), c).
What are coprime numbers?
Two or more numbers are coprime (or relatively prime) if their GCF is 1. Example: GCF(8, 15) = 1.
What are other names for GCF?
The GCF is also known as GCD (Greatest Common Divisor, math) and HCF (Highest Common Factor, British education). All three terms are equivalent.
Related Calculators
- LCM of 45 and 60 = 180
- Prime Factorization Calculator
- Divisors of a Number
- Multiples of a Number
- Fraction Calculator
- Fraction Simplifier (uses GCF to reduce fractions)
GCF Calculator
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