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Lowest Common Multiple and Highest Common Factor
Multiple:
The multiples of a number are its times table
If you want to find out if a number is a multiple of another number you just need to divide the first one by the second number, if the result is a round figure then the first number is a multiple of the second.
Any number can have an infinity of multiples
Example: 9 is a multiple of 3 because 9:3=3
144 is a multiple of 12 because 144:9=16 225 is a multiple of 5 because 225:5=45
In order to verify if a number is a multiple of another one here are some secret rules:
The number is divisible by:
2 if the last digit is 0, 2, 4, 6, or 8 (example:
12346);
3 if the sum of digits in the number are divisible
by 3
(example: 1236, because 1+2+3+6 = 12 = 3 x 4);
4 if the last 2 digits are divisible by 4
(example: 897544, because 44 = 4 x 11);
5 if the last digit is 0 or 5
(example: 178965 or 40980);
6 if it is divisible by 2 and 3;
7 sorry, no rule (you have to divide);
8 if the last 3 digits are divisible by 8
(example: 124987080, because 080 = 8 x 10;
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9 if the sum of digits is divisible by 9
(example: 234612, because 2+3+4+6+1+2 = 18 = 9 x 2);
10 if the last digit is 0
(example: 99990 );
100 if the last 2 digits are 0 (example 987600);
Lowest Common Multiple (LCM)
Definition: The Lowest Common Multiple of two numbers is the smallest number that can be divided by both numbers.
If you need to find the LCM of any numbers start by listing their multiples.
Find the common one.
Example:
1. The LCM of 10 and 12 is:
Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, Multiples of 12: 12, 24, 36, 48, 60, 72, 84,
2. The LCM of 14, 56 and 28 is:
Multiples of 56: 56, 112, 168, 224 Multiples of 28: 28, 56, 84, 112 Multiples of 14: 14, 28, 42, 56
Observation: Another way to find the LCM of two or three prime numbers you just need find the biggest power of each factor from the numbers and multiply these together
Example: The LCM of 50 and 20 is 50 = 2 * 5 * 5
20 = 2 * 2 * 5 LCM = 2* 2* 5* 5 =100
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Highest Common Factor (HCF)
The highest common factor of two or more whole numbers is the largest whole number that divides evenly into each of the numbers. There are two ways to find the highest common factor.
The first method is to list all of the factors of each number, then list the common factors and choose the largest one.
Example:
Find the HCF of 36 and 54.
The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
The factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54.
The common factors of 36 and 54 are 1, 2, 3, 6, 9, 18
Although the numbers in bold are all common factors of both 36 and 54, 18 is the highest common factor.
The second method for finding the greatest common factor is to list the prime factors, then multiply the common prime factors.
Example:
Let's use the same numbers, 36 and 54 again to find their greatest common multiple.
The prime factorization of 36 is 2 x 2 x 3 x 3 The prime factorization of 54 is 2 x 3 x 3 x 3
Notice that the prime factorizations of 36 and 54 both have one 2 andtwo 3s in common. So, we
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simply multiply these common prime factors to find the greatest common factor. Like this...
2 x 3 x 3 = 1
Other examples: The HCF of 50 and 20 is 50 = 2 * 5 * 5
20 = 2 * 2 * 5 HCF = 2* 5 = 10
The HCF of 70 and 120 is 70 = 2*5* 7
120 = 2*5* 3* 2 HCF = 2*5 = 10
See, both methods for finding the greatest common factor work!
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Let’s Practice:
1. Write this numbers as the product of their prime factors
a) 42 b) 64 c) 75 d) 125 e) 144 f) 269 g) 121
2. Find the lowest common multiple of the following.
a) 2, 3 b) 2, 3, 5 c) 7, 11 d) 11, 13
e) 3, 7, 2 3. Find the LCM of:
a) 12 and 24 b) 25 and 75 c) 13 and 39 d) 27 and 9 e) 60 and 10
4. Match the numbers with their LCM
9 and 13 84
7 and 12 117
14 and 20 45
15 and 45 140
5. What is the LCM for the given numbers?
a) 5, 7, 8 b) 8, 16, 32 c) 4, 12, 16
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d) 24, 36, 42
6. Find the first ten multiples of this numbers:
a) 12 b) 9 c) 15 d) 25 e) 30
7. Find the LCM of the next pairs of numbers a) 14 and 24
b) 80 and 25 c) 64 and 12 8. Find the right answer
The LCM of 45 and 20 is:
a) 180 b) 5 c) 60
The HCF of 30 and 50 is : a) 10
b) 150 c) 100
The HCF of 60 and 85 is:
a) 5 b) 15 c) 13
The LCM of 13, 26 and 169 a) 169
b) 13 c) 69 9. Let’s solve the table
16 and 18 is ____
10 and 40 is ____
30 and 10 is ____
6 and 4 is ____
9 and 21 is ____
7 9 and 6 is
____
4 and 18 is ____
9 and 18 is ____
40 and 10 is ____
30 and 15 is ____
45 and 35 is ____
12 and 18 is ____
18 and 24 is ____
24 and 27 is ____
16 and 18 is ____
45 and 25 is ____
16 and 14 is ____
15 and 20 is ____
24 and 21 is ____
12 and 6 is ____
18 and 15 is ____
6 and 15 is ____
15 and 10 is ____
24 and 18 is ____
8 and 10 is ____
12 and 14 is ____
27 and 15 is ____
14 and 10 is ____
21 and 18 is ____
15 and 6 is ____
40 and 10 is ____
15 and 18 is ____
12 and 9 is ____
21 and 9 is ____
12 and 24 is ____
24 and 6 is ____
18 and 27 is ____
35 and 25 is ____
15 and 27 is ____
18 and 15 is ____
20 and 15 is ____
25 and 45 is ____
8 and 14 is ____
9 and 6 is ____
10 and 18 is ____
16 and 10 is 15 and 27 is 40 and 35 is 12 and 27 is 40 and 20 is
____ ____ ____ ____ ____
10.Let’s solve the table
The Lowest Common Multiple (L.C.M.) of:
10 and 30 is ____
4 and 10 is ____
10 and 12 is ____
9 and 30 is ____
10 and 20 is ____
6 and 36 is ____
6 and 30 is ____
9 and 12 is ____
25 and 30 is ____
10 and 12 is ____
10 and 20 is ____
6 and 20 is ____
15 and 30 is ____
9 and 12 is ____
10 and 20 is ____
15 and 30 is ____
6 and 12 is ____
4 and 16 is ____
4 and 12 is ____
15 and 12 is ____
4 and 24 is ____
6 and 18 is ____
25 and 18 is ____
9 and 8 is ____
15 and 12 is ____
10 and 8 is ____
10 and 20 is ____
25 and 12 is ____
4 and 30 is ____
6 and 30 is ____
8 6 and 8 is
____
9 and 12 is ____
4 and 10 is ____
10 and 9 is ____
10 and 30 is ____
18 and 6 is ____
10 and 12 is ____
15 and 24 is ____
15 and 20 is ____
6 and 20 is ____
10 and 4 is ____
15 and 20 is ____
10 and 24 is ____
15 and 12 is ____
6 and 4 is ____
25 and 4 is ____
25 and 10 is ____
25 and 30 is ____
10 and 12 is ____
25 and 12 is ____
