In order to verify if a number is a multiple of
Lowest Common Multiple and Highest
another one here are some secret rules:
Common Factor
The number is divisible by:
Multiple:

 The multiples of a number are its times
2 if the last digit is 0, 2, 4, 6, or 8 (example:
12346);
table

 If you want to find out if a number is a
3 if the sum of digits in the number are divisible
by 3
multiple of another number you just
(example: 1236, because 1+2+3+6 = 12 = 3 x 4);
need to divide the first one by the

second number, if the result is a round
4 if the last 2 digits are divisible by 4
(example: 897544, because 44 = 4 x 11);
figure then the first number is a multiple

of the second.
5 if the last digit is 0 or 5
(example: 178965 or 40980);
 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

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;
225 is a multiple of 5 because 225:5=45
1

9 if the sum of digits is divisible by 9
Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80,
(example: 234612, because 2+3+4+6+1+2 = 18 =
Multiples of 12: 12, 24, 36, 48, 60, 72, 84,
9 x 2);


10 if the last digit is 0
2. The LCM of 14, 56 and 28 is:
(example: 99990 );
Multiples of 56: 56, 112, 168, 224
100 if the last 2 digits are 0
Multiples of 28: 28, 56, 84, 112
(example 987600);
Multiples of 14: 14, 28, 42, 56
Lowest Common Multiple (LCM)
Observation: Another way to find the LCM of two
or three prime numbers you just need find the
Definition: The Lowest Common Multiple of two
biggest power of each factor from the numbers and
numbers is the smallest number that can be divided
multiply these together
by both numbers.
Example: The LCM of 50 and 20 is
 If you need to find the LCM of any numbers
start by listing their multiples.
 Find the common one.
50
=
2*
20
=
2* 2*5
LCM =
Example:
1. The LCM of 10 and 12 is:
2
5*5
2* 2* 5* 5 =100
Highest Common Factor (HCF)
Although the numbers in bold are all common factors
of both 36 and 54, 18 is the highest common factor.
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
The second method for finding the greatest common
ways to find the highest common factor.
factor is to list the prime factors, then multiply the
The first method is to list all of the factors of each
common prime factors.
number, then list the common factors and choose
Example:
the largest one.
Let's use the same numbers, 36 and 54 again to find
Example:
their greatest common multiple.
Find the HCF of 36 and 54.
The prime factorization of 36 is 2 x 2 x 3 x 3
The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
The prime factorization of 54 is 2 x 3 x 3 x 3
The factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54.
Notice that the prime factorizations of 36 and 54 both
The common factors of 36 and 54 are 1, 2, 3, 6, 9, 18
have one 2 andtwo 3s in common. So, we
3
simply multiply these common prime factors to find
the greatest common factor. Like this...
2x3x3=1
Other examples: The HCF of 50 and 20 is
50
=
2*
20
=
2* 2*5
HCF =
2*
5*5
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!
4
e) 3, 7, 2
Let’s Practice:
3. Find the LCM of:
1. Write this numbers as
a)
12 and 24
the product of their
b)
25 and 75
prime factors
c)
13 and 39
a) 42
d)
27 and 9
b) 64
e)
60 and 10
c) 75
4. Match the numbers with their LCM
d) 125
e) 144
9 and 13
84
f) 269
7 and 12
117
g) 121
14 and 20
45
15 and 45
140
2. Find the lowest common multiple of the
5. What is the LCM for the given numbers?
following.
a) 2, 3
a) 5, 7, 8
b) 2, 3, 5
b) 8, 16, 32
c) 7, 11
d) 11, 13
c) 4, 12, 16
5
d) 24, 36, 42
The HCF of 30 and 50 is :
6. Find the first ten multiples of this numbers:
a) 10
a) 12
b) 150
b) 9
c) 100
c) 15
The HCF of 60 and 85 is:
d) 25
a) 5
e) 30
b) 15
7. Find the LCM of the next pairs of numbers
c) 13
a) 14 and 24
b) 80 and 25
The LCM of 13, 26 and 169
c) 64 and 12
a) 169
8. Find the right answer
b) 13
c) 69
The LCM of 45 and 20 is:
9. Let’s solve the table
a) 180
b) 5
c) 60
6
16 and 18 is 10 and 40 is 30 and 10 is 6 and 4 is
9 and 21 is
____
____
____
____
____
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
____
____
____
____
10.Let’s solve the table
____
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
____
____
____
____
The Lowest Common Multiple (L.C.M.) of:
____
____
____
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
____
____
____
____
9 and 6 is
10 and 18 is
____
____
____
____
9 and 30 is
10 and 20
is ____
____
is ____
____
is ____
6 and 36 is
6 and 30 is
9 and 12 is
25 and 30
10 and 12
____
____
____
is ____
is ____
10 and 20
6 and 20 is
15 and 30
9 and 12 is
10 and 20
is ____
____
is ____
____
is ____
15 and 30
6 and 12 is
4 and 16 is
4 and 12 is
15 and 12
is ____
____
____
____
is ____
4 and 24 is
6 and 18 is
25 and 18
9 and 8 is
15 and 12
____
____
is ____
____
is ____
10 and 8 is
10 and 20
25 and 12
4 and 30 is
6 and 30 is
____
is ____
is ____
____
____
____
20 and 15 is 25 and 45 is 8 and 14 is
____
10 and 12
____
40 and 10 is 15 and 18 is 12 and 9 is
____
4 and 10 is
____
12 and 14 is 27 and 15 is 14 and 10 is 21 and 18 is 15 and 6 is
____
10 and 30
16 and 10 is 15 and 27 is 40 and 35 is 12 and 27 is 40 and 20 is
7
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
15 and 24
15 and 20
6 and 20 is
____
is ____
is ____
is ____
____
10 and 4 is
15 and 20
10 and 24
15 and 12
6 and 4 is
____
is ____
is ____
is ____
____
25 and 4 is
25 and 10
25 and 30
10 and 12
25 and 12
____
is ____
is ____
is ____
is ____
8