Neft Teacher · Learn It

Unit 6 · Standard 6.NOS.4

Least Common Multiple

Lesson 6-12 · Learn It — how the math works

Why we're learning this

Astronaut Reyes checks the oxygen system every 4 hours. Astronaut Chen checks the water recycler every 6 hours. They both start their shifts at 6:00 AM. When is the next time both astronauts will be checking their systems at the same time?

✱ Math Workbench

Learn It

What is the least common multiple (LCM)?

A multiple is what you get when you count by a number: 4, 8, 12, 16. The least common multiple is the smallest multiple that two numbers share.

Picture itSee what the math looks like before you read about it

Two robotic arms on Station Helios run on repeating cycles. Arm A resets every 9 minutes and Arm B resets every 15 minutes. Right now both arms just reset at the same moment.
Where you see this in real lifeTwo robotic arms on Station Helios run on repeating cycles. Arm A resets every 9 minutes and Arm B resets every 15 minutes. Right now both arms just reset at the same moment.

Watch me solve itI do — read one step at a time

The problem looks like this
Tape diagram: multiples of 4 and multiples of 6 line up at 12 and 24, so the least common multiple of 4 and 6 is 12.24Multiples of 444444424Multiples of 66666
  1. I want the LCM of 4 and 6. I skip count the of 4: 4, 8, 12, 16, 20, 24.
  2. Then I skip count the of 6: 6, 12, 18, 24.
  3. I look for the first number in BOTH lists. 12 is in both lists, and it comes before 24.
  4. So the LCM of 4 and 6 is 12. I check: 12 ÷ 4 = 3 and 12 ÷ 6 = 2, so both go into 12 evenly.12 ÷ 4 = 3

Watch out — mistakes students make here

  • 3 is a common factor, not a multiple — you need a shared multiple.
  • 72 is a shared multiple, but look for a smaller one.
  • 30 minutes works for Rover A (30 = 2 × 15), but 30 is not a multiple of 20, so Rover B has not returned yet. The moment has to work for both rovers.

Try it with meWe do — you answer first, then check

Same steps, new numbers.

  1. Let's find the LCM of 3 and 5. What are the of 3?
    Check this step
    (3, 6, 9, 12, 15.)
  2. What are the of 5?
    Check this step
    (5, 10, 15.) What is the first number in both lists? (15.)
  3. So the LCM of 3 and 5 is 15.

On your ownYou do — solve this one by yourself

Before you start, remember:

  • Remember: the LCM is the SMALLEST the two numbers share.

Solve: Two rovers leave the space station at 8:00 AM. Rover A returns every 15 minutes and Rover B returns every 20 minutes. When is the next time both rovers are at the station together?

My answer

Stuck? Open one hint at a time.

Hint 1

Both rovers start together at 8:00 — they meet again after a number of minutes that both 15 and 20 divide evenly.

Hint 2

List each rover's return times in minutes and look for the first time that appears in both lists, then add it to 8:00 AM.

Hint 3

Rover A returns at 15, 30, 45, … minutes. Write Rover B's list (20, 40, …) underneath and find the first shared time, then add it to 8:00.

Check my answer
✅ Answer: 9:00 AM (60 minutes later)
LCM(15, 20) = 60 minutes. Both rovers return together 60 minutes after 8:00 AM, at 9:00 AM.

Turn & TalkSay it out loud with a partner

Astronaut Reyes checks oxygen every 4 hours and Chen checks water every 6 hours, both starting at 6:00 AM. Why isn't the next shared check simply 10 hours later (4 + 6)?

Try starting with:

  • Their checks line up at a ___ of both 4 and 6.Sus revisiones coinciden en un ___ de 4 y de 6.
  • Adding 4 + 6 is wrong because ___.Sumar 4 + 6 es incorrecto porque ___.
Use these wordsmultiplecommon multipleleast common multipleskip countingschedule