Common Multiples
4A · Chapter 3 · Lesson 2 · MATH-4A-C03-L02
Learning goal. Identify shared multiples of two numbers and explain the first common positive multiple.
Before you begin
Recall: Grade 3: arrays and multiplication tables; 4A Ch2: addition strategies; Multiples. Try these three warm-up problems. If they are difficult, revisit the prerequisite lesson before advancing.
Find the 5th positive multiple of 4.
Show answer & reasoning
4 × 5 = 20.
Is zero a multiple of 16? Explain.
Show answer & reasoning
0 = 16 × 0, so zero is a multiple of 16.
Find the 6th positive multiple of 11.
Show answer & reasoning
11 × 6 = 66.
Key vocabulary
- multiple
- whole-number product of a given number
- factor
- whole number dividing exactly
- prime
- positive number with exactly two positive factors
- composite
- positive number with more than two positive factors
Understand the concept
Common multiples occur on two skip-count lists. To find the least positive common multiple of 4 and 6, compare 4, 8, 12 and 6, 12: the first shared positive number is 12. We can also build equal-length cycles. Do not confuse the first common multiple with the greatest common factor.
Model it: paired skip-count lines with first intersection. Explain what every part of the model represents, then write a matching equation.
Fully worked examples
Read the problem first and predict the answer. Then study how each quantity is represented and calculated.
Worked example 1
Find the least positive common multiple of 15 and 14.
List positive multiples of each or use prime factors to locate the first shared positive multiple: 210.
Check your understanding 1
Find the least positive common multiple of 7 and 15.
Show answer & reasoning
List positive multiples of each or use prime factors to locate the first shared positive multiple: 105.
Worked example 2
Find the least positive common multiple of 3 and 13.
List positive multiples of each or use prime factors to locate the first shared positive multiple: 39.
Check your understanding 2
Find the least positive common multiple of 3 and 14.
Show answer & reasoning
List positive multiples of each or use prime factors to locate the first shared positive multiple: 42.
Worked example 3
Find the least positive common multiple of 15 and 7.
List positive multiples of each or use prime factors to locate the first shared positive multiple: 105.
Check your understanding 3
Find the least positive common multiple of 5 and 4.
Show answer & reasoning
List positive multiples of each or use prime factors to locate the first shared positive multiple: 20.
Guided practice
Find the least positive common multiple of 14 and 14.
Show answer & reasoning
List positive multiples of each or use prime factors to locate the first shared positive multiple: 14.
Find the least positive common multiple of 13 and 9.
Show answer & reasoning
List positive multiples of each or use prime factors to locate the first shared positive multiple: 117.
Find the least positive common multiple of 5 and 9.
Show answer & reasoning
List positive multiples of each or use prime factors to locate the first shared positive multiple: 45.
Find the least positive common multiple of 8 and 10.
Show answer & reasoning
List positive multiples of each or use prime factors to locate the first shared positive multiple: 40.
Will 126 be a common multiple of 9 and 7? Explain.
Show answer & reasoning
126 is two times their least common multiple, so both numbers divide it.
Independent practice
Find the least positive common multiple of 10 and 5.
Show answer & reasoning
List positive multiples of each or use prime factors to locate the first shared positive multiple: 10.
Find the least positive common multiple of 16 and 15.
Show answer & reasoning
List positive multiples of each or use prime factors to locate the first shared positive multiple: 240.
Find the least positive common multiple of 11 and 14.
Show answer & reasoning
List positive multiples of each or use prime factors to locate the first shared positive multiple: 154.
Will 140 be a common multiple of 10 and 7? Explain.
Show answer & reasoning
140 is two times their least common multiple, so both numbers divide it.
Will 176 be a common multiple of 8 and 11? Explain.
Show answer & reasoning
176 is two times their least common multiple, so both numbers divide it.
Find the least positive common multiple of 7 and 13.
Show answer & reasoning
List positive multiples of each or use prime factors to locate the first shared positive multiple: 91.
Find the least positive common multiple of 4 and 12.
Show answer & reasoning
List positive multiples of each or use prime factors to locate the first shared positive multiple: 12.
Find the least positive common multiple of 15 and 16.
Show answer & reasoning
List positive multiples of each or use prime factors to locate the first shared positive multiple: 240.
Challenge and transfer
Explain the relationships and justify your method. Multiple correct approaches may be accepted if the reasoning is valid.
A student says 48 is the least positive common multiple of 6 and 8 because both divide it. Find the actual least positive common multiple.
Show answer & reasoning
24 is the first positive multiple shared by 6 and 8; 48 is a later common multiple.
Find the least positive common multiple of 3 and 15.
Show answer & reasoning
List positive multiples of each or use prime factors to locate the first shared positive multiple: 15.
Exit check
Find the least positive common multiple of 4 and 13.
Show answer & reasoning
List positive multiples of each or use prime factors to locate the first shared positive multiple: 52.
Will 28 be a common multiple of 14 and 14? Explain.
Show answer & reasoning
28 is two times their least common multiple, so both numbers divide it.
Will 24 be a common multiple of 12 and 4? Explain.
Show answer & reasoning
24 is two times their least common multiple, so both numbers divide it.