1.12 HCF and LCM
For each of the following numbers:
List the factors.
State the highest common factor.
6 and 12
9 and 24
14 and 32
28 and 42
Consider the following prime factorisations:
180 = 2 \times 2 \times 3 \times 3 \times 5
600 = 2 \times 2 \times 2 \times 3 \times 5 \times 5
Find the highest common factor of 180 and 600.
Consider the following prime factorisations:
224 = 2 \times 2 \times 2 \times 2 \times 2 \times 7
196 = 2 \times 2 \times 7 \times 7
Find the highest common factor of 224 and 196.
Consider the following prime factorisations:
4900 = 2 \times 2 \times 5 \times 5 \times 7 \times 7
1750 = 2 \times 5 \times 5 \times 5 \times 7
Find the highest common factor of 4900 and 1750.
Consider the following prime factorisations:
140 = 2^{2} \times 5 \times 7
24\ 500 = 2^{2} \times 5^{3} \times 7^{2}
Find the highest common factor of 140 and 24\ 500.
Consider the following prime factorisations:
32\ 000 = 2^{8} \times 5^{3}
800 = 2^{5} \times 5^{2}
Find the highest common factor of 32\ 000 and 800.
Consider the numbers 234 and 108.
Write the prime factorisation of 234 in expanded form.
Write the prime factorisation of 108 in expanded form.
Find the highest common factor of 234 and 108.
Find the highest common factor of the following groups of numbers:
8 and 16
18 and 24
21 and 33
36 and 48
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 factorisations of several numbers in index form.
Find the highest common factor of 2940 and 14\ 700.
Find the highest common factor of 17\ 640 and 308\ 700.
Find the highest common factor of 2940,26\ 250, and 308\ 700.
| Number | Prime factorisation |
|---|---|
| 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} |
For each of the following numbers:
List the first 8 multiples.
State the lowest common multiple.
6 and 10
8 and 14
24 and 42
15 and 30
Consider the following prime factorisations:
54 = 2 \times 3 \times 3 \times 3
36 = 2 \times 2 \times 3 \times 3
Find the lowest common multiple of 54 and 36.
Consider the following prime factorisations:
2200 = 2 \times 2 \times 2 \times 5 \times 5 \times 11
2750 = 2 \times 5 \times 5 \times 5 \times 11
Find the lowest common multiple of 2200 and 2750.
Find the lowest common multiple of the following groups of numbers:
6 and 14
7 and 18
24 and 84
78 and 26
144 and 156
450 and 650
3, 8 and 18
9, 14 and 27
40, 260, and 140
128, 32, and 8
75, 125, and 225
16, 36, and 124
Consider the numbers 1575 and 1650.
Find the highest common factor of 1575 and 1650.
Find the lowest 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.
Allena has two pieces of cloth. One piece is 68 centimetres wide and the other piece is 32 centimetres wide. She wants to cut both pieces into strips, such that all of the strips have the same width.
How wide should she cut the strips if she wants the strips to be as large as possible?
How many strips will she end up with?
Mrs. Smith has three classes. Each class has 16, 28, and 36 students respectively. Mrs. Smith wants to divide each class into groups so that every group in every class has the same number of students and there are no students left over.
What is the maximum number of students Mrs. Smith can put into each group?
Jane has piece of cloth that has a size of 84 \text{ cm} by 112 \text{ cm}. She wants to cut squares of equal size out for a patchwork quilt without wasting any material. What would be the side length of the largest size square that can be cut?