GMAT Exam: A Simple Guide To Solving Word Problems - Greatest Common Factor and Least Common Multiple (GCF & LCM)

Greatest Common Factor
The greatest common factor (or GCF) of two numbers is the greatest factor common to both numbers. In other words, it is the largest number that evenly divides into each of the two numbers.

In GCF word problems you are seeking the greatest common number, the greatest common size, the greatest common measure, and so forth that can be used to evenly divide or distribute the objects or things from two unequal-size sets so that none are left over.


Example: A civic club has 24 boys and 36 girls as members. For a community project, the club’s faculty sponsor wants to evenly divide the boys and girls into groups, so that no one is left out. What is the greatest number of groups the faculty sponsor can make?
The greatest number of groups is the GCF (24, 36) = 12. Each of the 12 groups will have 2 boys and 3 girls in it.

Tip: Notice that 5 × 12 = 60, which is the total number of members (24 + 36 = 60).

Least Common Multiple
The least common multiple (or LCM) of two numbers is the least number that is a multiple of both numbers. In other words, it is the minimum number both numbers evenly divide into.

In LCM word problems you are seeking the minimum common number, the minimum common time, the minimum common measure, and so forth between two multiple events or items.


Example: At the entrance to an amusement park every 50th person gets a ticket for a free drink and every 75th person gets a ticket for a free ride. What is the minimum number of people who must enter for a person to receive both tickets?
The minimum number of people is the LCM (50, 75) = 150. The 150th person will be the first person to receive both tickets.

A quick way to compute the LCM of two numbers is to divide their product by their GCF.

For example,

See “Numbers and Operations” guide for additional discussion of greatest common factor and least common multiple.