Assignment #5.2: Greatest Common Factor and Least Common
Multiple
To find the greatest common factor (GCF), also sometimes called the greatest common
divisor, of two numbers, we look for the largest whole number that is a factor of both
numbers. This is sometimes called the greatest common divisor since it gives an integer
answer when divided into both numbers.
Example: Find the greatest common factor of 504 and 576
Solution: Put the prime factors of each number in a Venn diagram. The factors in the
overlapping part of the diagram will be the factors of the GCF, so multiply
them and find the solution. In this case, the GCF is 2 . 2 . 2 . 3 . 3 = 72.
504 = 2 . 2 . 2 . 3 . 3 . 7
576 = 2 . 2 . 2 . 2 . 2 . 2 . 3 . 3
To find the least common multiple (LCM) of two numbers, we look for the smallest
whole number that is divisible by both numbers.
Example: Find the least common multiple of 504 and 576
Solution: Put the prime factors of each number in a Venn diagram, as above. The factors
in the entire diagram will be the factors of the LCM, so multiply them and find the
solution. In this case, the LCM is 7 . 2 . 2 . 2 . 3 . 3 . 2 . 2 . 2 = 4032.
Example: Find the greatest common factor of 32 and 40.
Solution: This is a different approach to GCF problems. The technique is called “double
division” and it uses two series of divisions to discover common factors. At
each step the two numbers are divided by the same selected number, which
can be either prime or composite.
2 |32
2 |40
2 |16
2 |20
2 |8
2 |10
2 |4
|5
2
The GCF is composed of those factors that went into both original numbers.
In this example, it is the first three 2’s so the GCF is 2∙2∙2 = 8.
Example: Find the greatest common factor of 15 and 16
Solution: Put the prime factors of each number in a Venn diagram. These two numbers
have no factors in common, so the GCF is 1. Numbers such as these which do
not share any factors other than one are said to be relatively prime.
15 = 3 . 5
16 = 2 . 2 . 2 . 2
Example: Find the least common multiple of 25 and 150
Solution: Put the prime factors of each number in a Venn diagram. Since 150 is already
a multiple of 25, the LCM of 25 and 150 is 150.
25 = 5 . 5
150 = 2 . 3 . 5 . 5
5.2 Exercises:
Find the greatest common factor and least common multiple of each set of numbers. For
number 10, you may express your answer as a product of powers of primes. A Venn
diagram is not required.
1.
108 and 288
4.
450, 500 and
550
7.
28 and 225
2. 81 and 2000
3.
5. 1080 and 1800
6. 1800 and 180
8. 480 and 3456
9. 48, 72 and 288
10. 23 ∙ 34 ∙ 5 ∙ 72 ∙ 11 and 25 ∙ 3 ∙ 54 ∙ 7 ∙ 13
300, 400 and
700
11. p and q, if p and q are distinct prime numbers
12. Use the concept of GCF to answer the following question. What are the
dimensions of the largest square tile that can be used to pave a 30 ft. by 42 ft.
rectangular patio without any gaps or overlap? Use a sketch to show that your
answer is correct.
13. Use the concept of LCM to answer the following question. What are the
dimensions of the smallest square patio that can be paved by some number of 18
in. by 30 in. tiles without any gaps or overlap? Use a sketch to show that your
answer is correct.
14. What are all the values of n that make 546,324,16n divisible by 6?
15. If p and q are distinct odd primes, how many distinct factors does the number 2pq
have? For example, 1 and p are both factors. What are the others?
16. A garden has an area of 36 square meters and has sides that are integer lengths.
While it isn’t likely that the garden is 1 meter by 36 meters, that would have an
area of 36 square meters. What are all the other possible dimensions of this
garden?
17. List the prime numbers less than 100.
18. Twin primes are pairs of primes that are two apart. Examples include (3, 5) and
(5, 7). Find all other sets of twin primes which are less than 100.
19. For what single digit value of n is the number n5,3nn,672 divisible by 11?
20. *Container A is half full of water. Container B will hold 9 cups of water. When
the contents of container A is poured into container B, container B is 2/3 full.
How much water will container A hold?
21. #(c) How many different 5-card hands of poker can be dealt?
22. *In the Htam family, each daughter has the same number of brothers as she has
sisters, and each son has twice as many sisters as he has brothers. How many
sons and how many daughters are in the Htam family?
23. *A pile of pennies, dimes and half-dollars – 100 coins in all – is worth $5.00.
How many dimes are in the pile?