1. Question: If 5 is a factor of 35, what are all of the factors of 35?
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Question:
If 5 is a factor of 35, what are all of the factors of 35?
If 7 is a factor of 105, what are all of the factors of 105?
What is the smallest factor that 35 and 105 share? What is the largest?

2. 22 • 32 = 2 • 2 • 3 • 3 Use this to make an organized list of factors:
1 • 36;
2 • ____ 3 • ____ 4 (2•2) • ____
6 (2•3) • ____ 9 (3•3) • ____
12 (2•2•3) • ____ 18 (2•3•3) • ____
Notice the duplication.

3. Prime factorization for 72 = 23 • 32
How can you make sure you get every factor?
1, 2, 4, 8, 3, 9, 6, 12, 24, 18, 36, 72
Try this one: 60 = 22 • 3 • 5
1, 2, 4, 3, 5, 6, 10, 15, 12, 20, 60
Try this one: 48 = 24 • 3
1, 2, 4, 8, 16, 3, 6, 12, 24, 48

6. Now, look at multiples Suppose you have the number 20.
20 = 22 • 5. A multiple will be found by taking any whole number and multiplying it by 20 (22 • 5).
So: 3 • 20 = 60
So: 5 • 20 = 100
So: 8 • 20 = 160.

7. Let’s look at 2 numbers 120 84 2 • 60 2 • 42 2 • 30 2 • 21
2 • • 42
2 • • 21
2 • • 7
3 • 5

8. 120 = 2 • 2 • 2 • 3 • 5 84 = 2 • 2 • 3 • 7
So, 120 = 23 • 3 • 5 and 84 = 22 • 3 • 7
120: factors are 1, 2, 4, 8, 3, 5, 6, 12, 24, 10, 20, 40, 15, 30, 60, 120
84: factors are 1, 2, 4, 3, 7, 6, 12, 14, 28, 42, 84
To find their common factors, we see that they both have 1, 2, 4, 3, 6, and 12. The Greatest Common Factor is 12.
From the prime factorization: 2 • 2 • 3.

9. 120 = 2 • 2 • 2 • 3 • 5 84 = 2 • 2 • 3 • 7
120 : 120, 240, 360, 480, 600, 720, 840, 960,
84 : 84, 168, 252, 336, 420, 504, 588, 672, 756, 840, 924, 1008
840 = 120 • 7
840 = 84 • 10

10. 120 = 2 • 2 • 2 • 3 • 5 84 = 2 • 2 • 3 • 7
Now: multiples of 120 will be
2 • 2 • 2 • 3 • 5 (or 120) multiplied by some other number.
Multiples of 84 will be
2 • 2 • 3 • 7 (or 84) multiplied by some other number.
Begin with the factors of the larger number. Think: the LCM will be 120 multiplied by some other number. Underline the factors of the smaller number. Fill in the missing factors.
120 • 7 = 840.
840 = 7 • 120
840 = 10 • (10 is from 2 • 5)

11. Let’s try another together
Find LCM of 38 and 95
38 = 2 • = 5 • 19
LCM will be 38 multiplied by some number.
LCM will be 95 multiplied by some number

12. 38 : 76, 114, 152, 190, 228, 266, 304, 342, 380, 418, 456, 494, 532
95 : 95, 190, 285, 380, 475, 570, 665, 760, 855, 950
380 = 38 • 10
380 = 95 • 5

13. Another way to look at it
38 = 2 • 19
95 = 5 • 19
LCM = 19 • (what) is a multiple of 38 but not a factor of 95. 19

14. Let’s try a few more LCM of 28 and 140 126 = 2 • 3 • 3 • 7
140 = 2 • 2 • 5 • 7
Think: both 140 and 126 are factors of the LCM. Start with either factor:
126 (factors: a 2, two 3s, and a 7). From 140, we still need a 2 and a 5.
LCM = 126 • 2 • 5 = 1260.
Check: = 140 • 3 • 3 = 1260

15. Word Problem Example
Tara can run around the track in 5 minutes. Todd can run the same distance in 6 minutes, and Tony can do it in 8 minutes. If they start at the same time, when will they next meet at the starting line?
(Find the LCM of 5, 6, and 8)
Answer: 120 minutes or 2 hours

16. Homework--due Wednesday and Friday next week
Turn in completed Exploration 4.2 with tables 1 & 2 and Sieve (see online for details)
Section 4.3 p. 252 #1d, 2a, 3d, 5, 7, 10, 11, 14, 17ab (see online for details)