What Is the GCF of 17 and 31?
The GCF of 17 and 31 is 1. It is the biggest whole number that divides both 17 and 31 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 17 and 31
Write every factor of each number, then look for the ones in both lists.
- Factors of 17: 1, 17
- Factors of 31: 1, 31
- Common factors: 1
The only common factor is 1, so GCF(17, 31) = 1 and the two numbers are coprime.
Method 2: Use Prime Factorization
Break each number into primes. Keep the primes that appear in both, at the lowest power, and multiply them.
- 17 = 17
- 31 = 31
These two numbers share no primes at all, so there is nothing to multiply and the GCF is 1.
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.
- 31 = 1 × 17 + 14
- 17 = 1 × 14 + 3
- 14 = 4 × 3 + 2
- 3 = 1 × 2 + 1
- 2 = 2 × 1 + 0
The last non-zero remainder is 1, so the GCF of 17 and 31 is 1. All three methods agree.
Check the Answer
Divide each number by 1: 17 ÷ 1 = 17 and 31 ÷ 1 = 31. Both are whole numbers. Dividing by 1 changes nothing, which is exactly what a coprime pair should do.
Simplify the Fraction 17/31
Because the GCF is 1, 17/31 is already in its simplest form.
Related Answer: LCM of 17 and 31
Multiply the numbers and divide by the GCF: (17 × 31) ÷ 1 = 527 ÷ 1 = 527. Try your own numbers in the GCF Calculator.