The Factor Tree

Example 9 Find the least common multiple of 30 and 42.
Method 1 – Make a factor tree for each number.

Find the prime factors they have in common. (If you have three or more numbers,
your common factors need to appear in at least two of the numbers .)

In our problem, the numbers have a 2 and a 3 in common .
We will multiply the common factors, 2 and 3, along with any numbers that are
not in common, in this case 5, and 7. Our least common multiple (LCM) is:
LCM 2•3•5•7
LCM 210

Method 2 – Repeated division.
To do repeated division, we write our numbers in the manner below.

Now we want to find the smallest prime number that will divide evenly into one
or more of the numbers. Then divide and carry down the quotient. We repeat this
process until you are left with 1’s across the bottom. If you choose a prime
number that only divides evenly into one of the numbers , bring down the other
number.

To find the LCM, multiply the numbers to the left of the repeated division.

Example 10 Find the least common multiple of 12 and 54.

Example 11 Find the least common multiple of 4, 6, and 20.

Example 12 Find the least common multiple of 18, 36, and 81.

Note: To help you differentiate between the greatest common factor (GCF) and the least
common multiple (LCM) you can think that both include the common factors but the Least
common multiple also includes the Left over factors. (Least common multiple and Leftovers
both start with L.)

Example 13 Find the greatest common factor and least common multiple of 6 and
15.

Start by prime factoring each number. Write as a product of prime factors from least to greatest.
Greatest common factor: GCF = 3 Least common multiple: 3 is a common factor LCM 3•2•5 LCM 30	Identify and write the common factors. There is only one factor in common. The greatest common factor is that number. Identify the common prime factors. Multiply the common prime factors by any remaining prime factors.

Example 14 Find the greatest common factor and least common multiple of 8, 12,
and 21.
Start by prime factoring each number.

Write as a product of prime
factors from least to greatest.

2.1 EXERCISES

In 1-8, use a factor tree to write the prime factorization of each number.
1. 15
2. 12
3. 45
4. 80
5. 13
6. 240
7. 210
8. 1000

In 9-20, find the greatest common factor (GCF) of the fol lowing sets of numbers.
9. 8 and 12
10. 24 and 40
11. 12 and 35
12. 18 and 42
13. 9 and 10
14. 12 and 48
15. 120 and 216
16. 15, 20, and 30
17. 40, 50, and 60
18. 18, 30, and 42
19. 72, 108, and 180
20. 700, 420, and 1,120

In 21-32, find the least common multiple (LCM) of the following sets of numbers.
21. 3 and 4
22. 4 and 6
23. 25 and 80
24. 9 and 15
25. 12 and 20
26. 15 and 25
27. 4, 6, and 18
28. 3, 21, and 56
29. 70, 80, and 90
30. 10, 15, and 100
31. 7, 12, and 28
32. 11, 33, and 121

In 33-36, find the GCF and the LCM of the following sets of numbers.
33. 12 and 18
34. 45 and 60
35. 6, 8, and 24
36. 10, 25, and 30