Divisor
Also known as: factor
A divisor is the number you divide by in a division problem. More broadly, a divisor of an integer is any number that divides it evenly with no remainder — for example, 1, 2, 3, 4, 6, and 12 are the divisors of 12.
The word divisor has two closely related meanings. In any division problem — dividend ÷ divisor = quotient — the divisor is simply the number you divide by: in 20 ÷ 4 = 5, the divisor is 4. In number theory, a divisor of an integer n is any integer that divides n evenly, leaving no remainder. In this sense, divisor and factor mean the same thing.
To find the divisors of a number, look for pairs that multiply to it. The divisors of 12 are 1, 2, 3, 4, 6, and 12, coming from the pairs 1 × 12, 2 × 6, and 3 × 4. Every positive integer has 1 and itself as divisors; prime numbers have exactly those two and no others, which is what makes them prime.
Prime factorization gives a shortcut for counting divisors: write the number as a product of primes, add 1 to each exponent, and multiply. Since 12 = 2² × 3¹, it has (2 + 1)(1 + 1) = 6 divisors — matching the list above. Divisors also underpin the greatest common factor (the largest divisor two numbers share) and divisibility rules that let you test factors quickly.
Divisors and factors are core number theory content across many exams. The GRE tests divisors, prime factorization, and GCF/LCM in quantitative reasoning; the CLT covers factors, multiples, and number properties in math reasoning; and AMC competition problems lean on divisor-counting techniques and prime factorization regularly.
Key takeaways
- In a ÷ b, the divisor is b — the number you divide by.
- A divisor (or factor) of an integer divides it evenly with zero remainder.
- The divisors of 12 are 1, 2, 3, 4, 6, and 12; primes have exactly two divisors.
- Count a number's divisors from its prime factorization by adding 1 to each exponent and multiplying.
- The GRE, CLT, and AMC all test divisors, factors, and prime factorization.