GMAT Exam: A Simple Guide To Solving Word Problems - Divisors and Factors

Divisors and Factors

Divisibility Rules
Here are some useful divisibility rules to know for the GMAT.

- n, then it divides evenly into any multiple of n.

For instance, 3 divides 594, so it also divides (10)(594) = 5,940.
- m and n, then it divides evenly into am + bn, for any integers a and b.

For instance, 3 divides evenly into 27 and 3 divides evenly into 60, so 3 divides evenly into (5)(27) + 4(60) = 375.

A number’s factors divide evenly into the number.

Therefore, the results in this section concerning divisors and divisibility apply to factors as well.

Factors
If the prime factorization of a positive integer n is , where the p’s are distinct positive prime numbers and the k’s are their corresponding exponents, then the number of positive factors (or divisors) of n is the product (k1 + 1)(k2 + 1)…(kn + 1).

For example, the number of positive factors (or divisors) of n = a4b2c5d, where a, b, c, and d are prime numbers, is (4 + 1)(2 + 1)(5 + 1)(1 + 1) = (5)(3)(6)(2) = 180.
 

Remember, if no exponent is written on a variable or number, the exponent is understood to be 1 (for instance, d = d1).

On a smaller scale, the number of positive factors of 24, which equals (23)(3), is (3 + 1)(1 + 1) = (4)(2) = 8.

Because integers can have negative factors, the number of factors (positive and negative) of 24 is 2(8) = 16.

Knowing simple tricks about integers such as how to quickly determine the number of positive factors (or divisors) of a positive integer can turn what looks like a difficult question into an easy computation exercise.

See “Numbers and Operations” guide for additional discussion of prime numbers and factors.