What Is the GCF of 48 and 180?
The GCF of 48 and 180 is 12. It is the biggest whole number that divides both 48 and 180 with nothing left over. You will also see it called the HCF or the GCD. They are all the same number.
Method 1: List the Factors of 48 and 180
Write every factor of each number, then look for the ones in both lists.
- Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
- Factors of 180: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180
- Common factors: 1, 2, 3, 4, 6, 12
The biggest common factor is 12, so GCF(48, 180) = 12.
Method 2: Use Prime Factorization
Break each number into primes. Keep the primes that appear in both, at the lowest power, and multiply them.
- 48 = 2⁴ × 3
- 180 = 2² × 3² × 5
The shared primes at their lowest powers give 2² × 3 = 12. A prime that is missing from even one number never goes into the GCF.
Method 3: The Euclidean Algorithm
Divide the bigger number by the smaller, then divide the smaller by the remainder, and keep going until the remainder is 0. The last non-zero remainder is the GCF.
- 180 = 3 × 48 + 36
- 48 = 1 × 36 + 12
- 36 = 3 × 12 + 0
The last non-zero remainder is 12, so the GCF of 48 and 180 is 12. All three methods agree.
Check the Answer
Divide each number by 12: 48 ÷ 12 = 4 and 180 ÷ 12 = 15. Both are whole numbers. Now 4 and 15 have no common factor except 1, which proves that 12 was the greatest one.
Simplify the Fraction 48/180
Divide the top and bottom by the GCF: 48 ÷ 12 = 4 and 180 ÷ 12 = 15, so 48/180 = 4/15.
Related Answer: LCM of 48 and 180
Multiply the numbers and divide by the GCF: (48 × 180) ÷ 12 = 8,640 ÷ 12 = 720. Try your own numbers in the GCF Calculator.