What Is the GCF of 84 and 126?
The GCF of 84 and 126 is 42. It is the biggest whole number that divides both 84 and 126 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 84 and 126
Write every factor of each number, then look for the ones in both lists.
- Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
- Factors of 126: 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126
- Common factors: 1, 2, 3, 6, 7, 14, 21, 42
The biggest common factor is 42, so GCF(84, 126) = 42.
Method 2: Use Prime Factorization
Break each number into primes. Keep the primes that appear in both, at the lowest power, and multiply them.
- 84 = 2² × 3 × 7
- 126 = 2 × 3² × 7
The shared primes at their lowest powers give 2 × 3 × 7 = 42. 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.
- 126 = 1 × 84 + 42
- 84 = 2 × 42 + 0
The last non-zero remainder is 42, so the GCF of 84 and 126 is 42. All three methods agree.
Check the Answer
Divide each number by 42: 84 ÷ 42 = 2 and 126 ÷ 42 = 3. Both are whole numbers. Now 2 and 3 have no common factor except 1, which proves that 42 was the greatest one.
Simplify the Fraction 84/126
Divide the top and bottom by the GCF: 84 ÷ 42 = 2 and 126 ÷ 42 = 3, so 84/126 = 2/3.
Related Answer: LCM of 84 and 126
Multiply the numbers and divide by the GCF: (84 × 126) ÷ 42 = 10,584 ÷ 42 = 252. Try your own numbers in the GCF Calculator.