6 — Multiples and Common Multiples

Multiples and Common Multiples

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Multiples are like skip counting - they're what you get when you multiply a number by 1, 2, 3, 4, and so on! While factors help us break numbers apart, multiples help us build them up. Understanding multiples is super important because we'll use them later when working with fractions.

Tip

Key Rule: To find multiples of any number, just multiply it by 1, 2, 3, 4, 5... and keep going! The multiples go on forever.

Multiples and Common Multiples · 1:46

Worked Example

Problem

Find the first six multiples of 7.

Solution

7, 14, 21, 28, 35, 42

Explanation

  1. 1Multiply 7 by 1: 7 × 1 = 7.
  2. 2Multiply 7 by 2: 7 × 2 = 14.
  3. 3Multiply 7 by 3: 7 × 3 = 21.
  4. 4Multiply 7 by 4: 7 × 4 = 28.
  5. 5Multiply 7 by 5: 7 × 5 = 35.
  6. 6Multiply 7 by 6: 7 × 6 = 42. So the first six multiples are 7, 14, 21, 28, 35, 42.

Worked Example

Problem

Find the LCM of 6 and 9.

Solution

18

Explanation

  1. 1List multiples of 6: 6, 12, 18, 24, 30, 36...
  2. 2List multiples of 9: 9, 18, 27, 36, 45...
  3. 3Look for the smallest number that appears in both lists.
  4. 4The number 18 appears in both lists first, so 18 is the LCM.
  5. 5Check: 18 ÷ 6 = 3 and 18 ÷ 9 = 2, so 18 works!

Question

List the first four multiples of 5.

Show Answer
5, 10, 15, 20
Show Solution
  1. 1Multiply 5 by 1: 5 × 1 = 5.
  2. 2Multiply 5 by 2: 5 × 2 = 10.
  3. 3Multiply 5 by 3: 5 × 3 = 15.
  4. 4Multiply 5 by 4: 5 × 4 = 20. So the first four multiples of 5 are 5, 10, 15, 20.

Question

Is 42 a multiple of 7? Explain your answer.

Show Answer
Yes, 42 is a multiple of 7 because 7 × 6 = 42
Show Solution
  1. 1To check if 42 is a multiple of 7, we need to see if 7 divides evenly into 42.
  2. 2We can divide: 42 ÷ 7 = 6.
  3. 3Since we get a whole number (6), this means 7 × 6 = 42.
  4. 4Because 42 equals 7 times a whole number (6), 42 is indeed a multiple of 7.

Question

Find two common multiples of 4 and 6.

Show Answer
12 and 24 (other correct answers: 36, 48, etc.)
Show Solution
  1. 1List multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48...
  2. 2List multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48...
  3. 3Look for numbers that appear in both lists.
  4. 4The numbers 12, 24, 36, and 48 all appear in both lists.
  5. 5Any two of these work as common multiples. The most common answers are 12 and 24.

Question

Find the LCM of 9 and 15.

Show Answer
45
Show Solution
  1. 1List multiples of 9: 9, 18, 27, 36, 45, 54...
  2. 2List multiples of 15: 15, 30, 45, 60...
  3. 3Find the smallest number that appears in both lists.
  4. 4The number 45 is the first number that appears in both lists.
  5. 5Check our answer: 45 ÷ 9 = 5 and 45 ÷ 15 = 3, so 45 is divisible by both numbers. Therefore, the LCM is 45.

Question

A bakery makes cookies in batches of 8 and muffins in batches of 12. If they want to package equal numbers of cookies and muffins together, what is the smallest number of each they need to make?

Show Answer
24 cookies and 24 muffins
Show Solution
  1. 1This problem is asking for the LCM of 8 and 12.
  2. 2List multiples of 8: 8, 16, 24, 32, 40...
  3. 3List multiples of 12: 12, 24, 36, 48...
  4. 4Find the smallest common multiple: 24 appears in both lists first.
  5. 5This means they need 24 cookies (3 batches of 8) and 24 muffins (2 batches of 12) to have equal amounts of each.

Tip

Remember: Multiples help us skip count and find common denominators. The LCM is the smallest number that both original numbers divide into evenly - it's like finding the perfect meeting point for two different patterns!