What Is the GCF of 45 and 90?
The greatest common factor (GCF) of 45 and 90 is 45. Because 45 divides 90 evenly, the GCF equals the smaller number - no further calculation needed. Use the calculator below for step-by-step work, or simplify 45/90 to 1/2 by dividing both by 45.
Enter whole numbers
Type two or more integers separated by commas or spaces.
Result
Greatest common factor (GCF), also called HCF or GCD.
GCF of 45, 90
45
Simplified: 45/90 → 1/2
Show work
Step 1: Factors of 45
[1], [3], [5], [9], [15], [45]
Step 2: Factors of 90
[1], 2, [3], [5], 6, [9], 10, [15], 18, 30, [45], 90
Common factors (in every list)
1, 3, 5, 9, 15, 45
Greatest common factor = 45
Methodology and limitations
Last reviewed:
Methodology
Computes the greatest common factor using the Euclidean algorithm for the final answer, with optional step-by-step work via listing factors, prime factorization, or repeated division with remainders.
Limitations
Inputs must be whole numbers within JavaScript safe-integer range. GCF(0, 0) is undefined. Step-by-step listing and prime factorization are hidden for very large values to keep the page responsive.
The GCF of 45 and 90 Is 45
The greatest common factor of 45 and 90 is 45 - the largest whole number that divides both values evenly. Common factors shared by 45 and 90 are 1, 3, 5, 9, 15, 45; 45 is the greatest among them. Because 45 divides 90 with no remainder, the GCF equals the smaller number.
When One Number Divides the Other
90 ÷ 45 = 2 with no remainder, so every factor of 45 is also a factor of 90. That makes 45 the greatest common factor - you do not need to list all factors or run the Euclidean algorithm to confirm GCF(45, 90) = 45.
Listing Factors of 45 and 90
Factors of 45: 1, 3, 5, 9, 15, 45. Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90. The numbers that appear in both lists are 1, 3, 5, 9, 15, 45, so the greatest common factor is 45.
Prime Factorization of 45 and 90
45 = 3² × 5. 90 = 2 × 3² × 5. Multiply the shared prime powers (3² × 5) to get GCF = 45.
Simplify 45/90 Using the GCF
Divide numerator and denominator by 45: 45 ÷ 45 = 1 and 90 ÷ 45 = 2. So 45/90 in lowest terms is 1/2.
Common Factors of 45 and 90
| Factor | Divides 45? | Divides 90? |
|---|---|---|
| 1 | Yes | Yes |
| 3 | Yes | Yes |
| 5 | Yes | Yes |
| 9 | Yes | Yes |
| 15 | Yes | Yes |
| 45 (GCF) | Yes | Yes |
Frequently Asked Questions
What is the GCF of 45 and 90?
The GCF of 45 and 90 is 45. It is the largest positive integer that divides both 45 and 90 evenly with no remainder. Common factors shared by both numbers are 1, 3, 5, 9, 15, 45; 45 is the greatest among them.
How do you find the GCF of 45 and 90?
List the factors of 45 and 90, then pick the largest number in both lists - that is 45. You can also use prime factorization: multiply the shared prime powers, or apply the Euclidean algorithm by dividing the larger number by the smaller until the remainder is zero. All three methods are shown in the calculator above.
What are the common factors of 45 and 90?
The common factors of 45 and 90 are 1, 3, 5, 9, 15, 45. These are the whole numbers that divide both 45 and 90 without a remainder. The greatest among them is 45, which is the GCF.
How do you simplify 45/90 using the GCF?
Divide the numerator and denominator of 45/90 by their GCF (45). 45 ÷ 45 = 1 and 90 ÷ 45 = 2, so 45/90 simplifies to 1/2 in lowest terms.
Is the GCF of 45 and 90 the same as the GCD?
Yes. GCF (greatest common factor), HCF (highest common factor), and GCD (greatest common divisor) all name the same value for 45 and 90: 45. The terms differ by context - GCF in grade-school math, GCD in number theory - but the calculation is identical.
Why is the GCF of 45 and 90 equal to 45?
When one number is a multiple of the other, the smaller number divides the larger evenly. Every divisor of the smaller number also divides the larger, so the greatest shared divisor is the smaller value itself. Here, 45 divides 90, so GCF(45, 90) = 45.