1.07 GCF and LCM using prime factorizations
Consider the following prime factorizations:
180 = 2 \times 2 \times 3 \times 3 \times 5
600 = 2 \times 2 \times 2 \times 3 \times 5 \times 5
Find the greatest common factor of 180 and 600.
Consider the following prime factorizations:
224 = 2 \times 2 \times 2 \times 2 \times 2 \times 7
196 = 2 \times 2 \times 7 \times 7
Find the greatest common factor of 224 and 196.
Consider the following prime factorizations:
4900 = 2 \times 2 \times 5 \times 5 \times 7 \times 7
1750 = 2 \times 5 \times 5 \times 5 \times 7
Find the greatest common factor of 4900 and 1750.
Consider the following prime factorizations:
140 = 2^{2} \times 5 \times 7
24\ 500 = 2^{2} \times 5^{3} \times 7^{2}
Find the greatest common factor of 140 and 24\ 500.
Consider the following prime factorizations:
32\ 000 = 2^{8} \times 5^{3}
800 = 2^{5} \times 5^{2}
Find the greatest common factor of 32\ 000 and 800.
Consider the numbers 234 and 108.
Write the prime factorization of 234 in expanded form.
Write the prime factorization of 108 in expanded form.
Find the greatest common factor of 234 and 108.
Find the greatest common factor of the following groups of numbers:
42 and 24
72 and 108
121 and 143
150 and 560
48, 80, and 176
168, 312, and 120
168, 224, and 280
150, 450, and 375
The table below shows the prime factorizations of several numbers in exponential form:
Find the greatest common factor of 2940 and 14\ 700.
Find the greatest common factor of 17\ 640 and 308\ 700.
Find the greatest common factor of 2940, 26\ 250, and 308\ 700.
| Number | Prime factorization |
|---|---|
| 2940 | 2^{2} \times 3 \times 5 \times 7^{2} |
| 14\ 700 | 2^{2} \times 3 \times 5^{2} \times 7^{2} |
| 17\ 640 | 2^{3} \times 3^{2} \times 5 \times 7^{2} |
| 26\ 250 | 2 \times 3 \times 5^{4} \times 7 |
| 308\ 700 | 2^{2} \times 3^{2} \times 5^{2} \times 7^{3} |
Consider the following prime factorizations:
54 = 2 \times 3 \times 3 \times 3
36 = 2 \times 2 \times 3 \times 3
Find the least common multiple of 54 and 36.
Consider the following prime factorizations:
2200 = 2 \times 2 \times 2 \times 5 \times 5 \times 11
2750 = 2 \times 5 \times 5 \times 5 \times 11
Find the least common multiple of 2200 and 2750.
Find the least common multiple of the following groups of numbers:
24 and 84
78 and 26
144 and 156
450 and 650
40, 260, and 140
128, 32, and 8
75, 125, and 225
16, 36, and 124
Consider the numbers 1575 and 1650.
Find the greatest common factor of 1575 and 1650.
Find the least common multiple of 1575 and 1650.
Cicada species spend many years underground, emerging as a single brood every generation. Magicicada cassinii emerges every 17 years. They are hunted by the Latrodectus mactans spider, which has a 3 year breeding cycle.
How often do these species start their breeding cycle at the same time?
A genetic mutation changes the life cycle of the cicadas so the mutants emerge every 18years. How often will the mutant cicada species start their life cycle at the same time as the spiders?
Two airlines, JetFuel and LionAir, each land planes at an airport terminal. During the day a JetFuel plane lands every 18 minutes, and a LionAir plane lands every 16 minutes.
A shuttlebus takes passengers from the terminal to the city center, departing every 21 minutes.
How often does a JetFuel plane land and a shuttlebus depart at the same time?
Which of these two options happens more often in a single day?
JetFuel plane lands and the shuttlebus departs at the same time.
LionAir plane lands and the shuttlebus departs at the same time.