Least Common Multiple Word Problems

Concept

The abbreviation LCM stands for Least Common Multiple. The least common multiple of a number is the smallest number that is the product of two or more numbers. The least common multiple can be calculated for two or more integers as well as two or more fractions. The least common multiple of two numbers is the lowest possible number that can be divisible by both numbers.

There is more than one method to find the LCM of two or more numbers.
1. Listing Method
2. Division/Prime Factorization Method

Rules

To find the LCM using listing method:
1. List the multiples of the numbers that you have to find the LCM of.
2. Find the first common product that appears in both lists.

To find the LCM using division/prime factorization method:
1. Find the least prime that one or more of the numbers are divisible by.
2. Divide the number or numbers by the primes, from the least prime to the greatest prime, until the numbers are reduced to 1. (If a number is not divisible by the prime, bring it down.)
3. Multiply all the primes that were used in this process. The answer is the LCM.

Example

An airline stewardess flies to Paris every 10 days for one day. Another stewardess flies to Paris every 15 days for one day. Today, both are in Paris. After how many days will both stewardesses be in Paris again on the same day?

Solution

Method 1 (Listing Method)
Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100,…
Multiples of 15: 15, 30, 45, 60, 75, 90, 105,…
LCM of 10 and 15 is 30.

Method 2 (Division/Prime Factorization)

The LCM = 2 × 3 × 5 = 30
After 30 days both stewardesses will be in Paris again on the same day.

Practice Least Common Multiple Word Problems

Practice Problem 1

A salesman goes to New York every 15 days for one day and another every 24 days, also for one day. Today, both are in New York. After how many days both salesmen will be again in New York on the same day?

Practice Problem 2

Today, both the soccer team and the basketball team had games. The soccer team plays every three days, and the basketball team plays every five days. When will both teams have games on the same day again?

Practice Problem 3

Sam can save 9 dollars every day while Maria can save 12 dollars. What is the least number of days it will take each person to save the same amount of money?