Understanding Least Common Multiple (LCM) and Greatest Common Factor (GCF) is crucial for success on the ACT® Math section.
These concepts not only appear in direct questions but also in word problems and fraction-related tasks. Let's break down what LCM and GCF mean and how they show up on the test.
What are LCM and GCF?
LCM (Least Common Multiple)
The smallest number that's a multiple of two or more numbers. It's like finding the first time multiple number lines would meet up.
Example with 3 and 4:
- Multiples of 3: 3, 6, 9, 12, ...
- Multiples of 4: 4, 8, 12, ...
The LCM of 3 and 4 is 12, as it's the smallest number that appears in both lists.
GCF (Greatest Common Factor)
The largest number that's a factor of two or more numbers. Think of it as the biggest number that can divide evenly into all the given numbers.
We use prime factorization to get factors. Prime is any number only divisible by 1 and itself. We do this to find the factors of each term in the lowest possible terms.
Example with 6 and 9:
- 6: 3 · 2
- 9: 3 · 3
The largest factor both share is 3. Therefore, the GCF is 3.
The GCF can help us determine how to simplify a fraction. The LCM can help us determine the least common multiple to put the denominator of a fraction.
LCM and GCF on the ACT® Math
On the ACT® Math, you might see LCM and GCF pop up in three main ways:
- In fraction problems (often disguised!)
- Direct questions asking for LCM or GCF
- Word problems about scheduling (LCM) or divisibility (GCF)
Questions on LCM/GCF don't show up frequently. It's important to understand how to do these questions at a basic level. Then I'll show you a really easy way to solve them below.
Practice Section
Let's tackle some ACT®-style questions to see how LCM and GCF work in real test scenarios.
Question:
You are coordinating schedules for two employees at a store. Aria's shift repeats every 4 days, while Jamal's shift repeats every 6 days. If both employees start their shifts on the same day, after how many days will both employees be scheduled to work together again?
- 8 days
- 12 days
- 16 days
- 18 days
- 24 days
Click for the Answer
Correct Answer: B. 12 days
Explanation:
To solve this problem, we need to find the Least Common Multiple (LCM) of 4 and 6. The LCM of two numbers is the smallest number that is a multiple of both.
- List the multiples of each number:
- Multiples of 4: 4, 8, 12, 16, 20, 24, ...
- Multiples of 6: 6, 12, 18, 24, ...
- Find the smallest common multiple: 12
This means both employees will be scheduled to work together again after 12 days.
Calculator Tip
If you have a graphing calculator, you can type in GCF/LCM to solve the math for you.
Here's the way to do it on a Texas T-83/84® calculator:
For LCM:
- Press [MATH] then [Right Arrow] to select "NUM".
- Press [8] to select the 8: lcm( function.
For GCD (another term for GCF):
- Press [MATH] -> Num [9].
LCM Function on TI-83/84
GCD Function on TI-83/84
Conclusion
LCM and GCF are important concepts that can appear in various forms on the ACT® Math. By understanding these principles and practicing with different question types, you'll be better prepared to tackle these problems efficiently on test day.
Remember to use your calculator when appropriate, and look for opportunities to apply LCM and GCF concepts, even when they're not explicitly mentioned in the question.