H.C.F. and L.C.M. 2 (Common Multiples, L.C.M.) NOTES
Level 1:
1
Find all the common multiples of 6 and 8 up to 50? How many common multiples are there?
Step 1:
List all the multiples of 6 up to 50:
6, 12, 18, 24, 30, 36, 42, 48
Step 2:
List all the multiples of 8 up to 50:
8, 16, 24, 32, 40, 48
Step 3:
The common multiples of 6 and 8 up to
24, 48
50:
Step 4:
There are
2
common multiples.
Level 2:
1
Use listing method to find the L.C.M. (Least Common Multiple) of 10 and 25.
Step 1:
List the multiples of 10:
10, 20, 30, 40, 50, 60, 70, 80, 90, 100 ….
Step 2:
List the multiples of 25:
25, 50, 75, 100, 125, 150 …
Step 3:
The common multiples of 10 and 25:
50, 100 …
Step 4:
The L.C.M. (Least Common Multiple) of 10 and
50
25:
2
Use listing method to find the L.C.M. (Least Common Multiple) of 7 and 14.
Step 1:
List the multiples of 7:
7, 14, 21, 28, 35, 42, 49, 56, 70 …
Step 2:
List the multiples of 14:
14, 28, 42, 56, 70, 84 …
Step 3:
The common multiples of 7 and 14:
14, 28, 42, 56, 70 …
Step 4:
The L.C.M. (Least Common Multiple) of 7 and 14:
14
Finding:
7 X 2 = 14
14 is a multiple of 7.
14 is a multiple of 7. It is also the L.C.M. of 7 and 14.
If one of the numbers is a multiple of the other one, it is also their L.C.M..
3
Use the short division to find the L.C.M. of 12 and 18.
common
3
1
2
,
4
,
1
8
Step 1: Divide the two numbers by a common factor.
6
Step 2: Repeat the steps until there are no more common factors
factor
common
2
apart from 1.
factor
2
2
1
L.C.M. of 12 and 18
=
=
3X2X2X3
36
3
,
3
Step 3: Multiply all the common factors on the left column and the
numbers left in the bottom row.
4
Use the short division to find the L.C.M. of 10, 12 and 15.
common factor
2
1
0
1
2
,
1
5
common factor
3
5
6
1
5
5
5
2
5
1
2
1
Step 1: Divide the three numbers by a common factor.
Step 2: Repeat the steps until there are no more common
factors apart from 1.
**If one of the numbers is not divided exactly, we bring it
down in the next row.
Step 3: Multiply all the common factors on the left column.
L.C.M. of 10, 12 and 15:
=
=
5
2X3X5X1X2X1
60
Find the L.C.M. of 9 and 13.
Findings:
9 and 13 do not have common factors except 1.
Step 1:
Multiply the pair of numbers.
common factor
1 9
1 3
9
1 3
,
The product is the Least Common Multiple (L.C.M.)
L.C.M. of 9 and 13.
=
9 X 13
=
117
When the pair of numbers has no common factors except 1, their product is the L.C.M.
The product of the pair of numbers MUST be their common multiple, but MAY NOT be their L.C.M.
Level 3:
1a
Use the listing method to find the L.C.M. of 4 and 6.
Step 1:
List the multiples of 4:
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60 …
Step 2:
List the multiples of 6;
6, 12, 18, 24, 30, 36, 42, 48, 54, 60 …
Step 3:
The common multiples of 4 and 6:
12, 24, 36, 48, 60 …
Step 4:
The L.C.M. of 4 and 6:
12
Finding:
The multiples of 12
= 12, 24, 36, 48, 60 ….
= the common multiples of 4 and 6
The common multiples of the numbers are the multiples of their L.C.M.
Refer to Level 1 questions, use a faster method to find out the answers.
1b
Find all the common multiples of 6 and 8 up to 50? How many common multiples are there?
Step 1:
L.C.M of 6 and 8
= 24
Step 2:
The multiples of L.C.M. = 24 , 48, 72 …
Step 3:
∵ The common multiples of the numbers are the multiples of their L.C.M.
∴ The common multiples of 6 and 8 up to 50 are
There are
2
2
common multiples.
What is the L.C.M if the fourth common multiple of 2 and 7 is 56?
Step 1:
th
4 common multiple = 56
st
1 common multiple = L.C.M of 2 and 7
= 56 ÷ 4
= 14
The L.C.M of 2 and 7 is
14
.
24, 48
.
3
What is the third comment multiple of 6 and 10?
Step 1:
∵ the common multiples of the numbers are the multiples of their L.C.M.
rd
∴ the 3 common multiple of 6 and 10
rd
= the 3 common multiple of L.C.M. of 6 and 10.
Step 2:
2 6
L.C.M of 6 and 10
1 0
3
5
= 30
Step 3:
st
1 common multiple of 6 and 10 = L.C.M of 6 and 10 = 30
nd
2 common multiple of 6 and 10 = L.C.M X 2 = 30 X 2 = 60
rd
3 common multiple of 6 and 10 = L.C.M. X 3 = 30 X 3 = 90
The third common multiple of 6 and 10 is
4
90
.
Find any two numbers that have 28 as their L.C.M.
Hint:
1: If one of the numbers is a multiple of another one, it is also their L.C.M.
∴ 28 must be one of the numbers and must be the multiple of another number.
2: When the pair of numbers has no common factors except 1, their product is the L.C.M.
∴ A X B = 28
The two numbers can be
28 and 1
5
,
28 and 2
,
28 and 4
,
4 and 7
,
28 and 7
A is a multiple of B. If A is 50, what is the L.CM of A and B?
Hint:
If one of the numbers is a multiple of another one, it is also their L.C.M.
If A is a multiple of B, A is the L.C.M of A and B
The L.C.M of A and B is
50
,
28 and 14
.
6
Find the common multiple of 11 and 33 that is nearest to 200 without using listing method.
Step 1:
∵ 33 is a multiple of 11, it is also their L.C.M..
L.C.M of 11 and 33
= 33
Step 2:
33 X ? = 200 Or nearest to 200.
? = 200 ÷ 33
?=6…2
Step 3:
33 X 6 = 198
 nearest to 200
∵ the common multiples of the numbers are the multiples of
their L.C.M,
33 X 7 = 231
th
∴ the 6 common multiple of 11 and 33 = L.C.M X 6
The common multiple of 11 and 33 that is nearest to 200 is
231
.
Level 4:
1
Find the smallest number that can be divided by both 7 and 13 without any remainders.
Hints:
1: Smallest  Least Common Multiple
2: divided by  the answer (the dividend) should be larger or same as 7 and 13
 look for a multiple, not a factor
∴ we need to look for L.C.M of 7 and 13.
Step 1:
∵ 7 and 13 have no common factors except 1, their product is the L.C.M.
L.C.M of 7 and 13
= 7 X 13
= 91
The smallest number that can be divided by both 7 and 13 without any remainders is
Checking:
91 ÷ 7 = 13 √
91 ÷ 13 = 7 √
91
.
2
Find the smallest number which on adding 19 to it is exactly divisible by 28, 36 and 45.
Hints:
1: Smallest  Least Common Multiple
2: divisible by  the answer (the dividend) should be larger or same as 28, 36 and 45.
 look for a multiple, not a factor
∴ we need to look for L.C.M .
Step 1:
L.C.M of 28, 36 and 45
=9X4X7X1X5
9 2 8
3 6
4
4 2 8
4
5
7
1
5
5
= 1260
Step 2:
(? + 19) ÷ 28 = Quotient without remainder
(? + 19) = 1260
? = 1260 – 19
? = 1241
Checking:
The smallest number is
(1241 +19) ÷ 28
= 1260 ÷ 28
= 45
√
(1241 +19) ÷ 36
= 1260 ÷ 36
= 35
√
(1241 +19) ÷ 45
= 1260 ÷ 45
= 28
√
1241
.
3
Find the smallest number which subtracting 15 from it is exactly divisible by 8, 12 and 15.
Hints:
1: Smallest  Least Common Multiple
2: divisible by  the answer (the dividend) should be larger or same as 8, 12 and 15.
 look for a multiple, not a factor
∴ we need to look for L.C.M .
Step 1:
L.C.M of 8, 12 and 15
=4X3X2X1X5
4 8
1 2
1 5
3 2
3
1 5
2
1
5
= 120
Step 2:
(? – 15) ÷ 8 = Quotient without remainder
(? – 15) = 120
? = 120 + 15
? = 135
Checking:
The smallest number is
(135 – 15) ÷ 8
= 120 ÷ 8
= 15
√
(135 – 15) ÷ 12
= 120 ÷ 12
= 10
√
(135 – 15) ÷ 15
= 120 ÷ 15
=8
√
135
.
4
A bag of marbles can be evenly distributed among 16 or 24 children. What is the smallest number of marbles in the
bag? How many marbles does each child get?
Hints:
1: Smallest  Least Common Multiple
2: distributed among  the answer (the dividend) should be larger or same as 16 or 24.
 look for a multiple, not a factor
∴ we need to look for L.C.M. of 16 and 24.
Step 1:
8 1 6
L.C.M. of 16 and 24.
2
=8X2X3
= 48
The smallest number of marbles in the bag is
48
.
48 ÷ 16
=3
If there are 16 children, each child gets
3
marbles.
2
marbles.
48 ÷ 24
=2
If there are 24 children, each child gets
2 4
3
5
Sam goes for a walk every 2 days. Paul goes for a walk every 4 days. They meet in the park on a Sunday. At least
how many days later will they meet again? On which day will they meet again?
Hints:
1: The days Sam goes for a walk multiples of 2
e.g. If Sam goes to the park every 2 days, he will go to the park again 2 days, 4 days, 6 days….
2: The days Paul goes for a walk  multiples of 4
e.g. If Paul goes to the park every 4 days, he will go to the park again 4 days, 8 days, 12 days….
3: at least  Least Common Multiple
∴ we are looking for L.C.M of 2 and 4.
Step 1:
L.C.M. of 2 and 4
Since 4 is a multiple of 2, it is also their L.C.M..
=4
They will meet again at least
Sun
st
1 meet
4
days later.
Mon
Tue
Wed
Thurs
1 day
2 days
3 days
4 days
(S)(P)
nd
2 meet
(S)
(S)(P)
S = Sam, P = Peter
They will meet again on
Thursday
.
Fri
Sat
6
Buses leave a station every 18 minutes and minibuses leave the same station every 24 minutes. At eleven o’clock in
the morning, a bus and a minibus leave the station at the same time. After At least how many minutes will a bus and
a minibus leave at the same time again? What time will a bus and a minibus leave at the same time again?
Hints:
1: Buses leave a station every 18 minutes multiples of 18
e.g. If buses leave every 18 minutes, it will leave again 18 minutes, 36 minutes, 54 minutes…
2: Minibuses leave a station every 24 minutes  multiples of 24
e.g. If minibuses leave every 24 minutes, it will leave again 24 minutes, 48 minutes, 72 minutes…
3: at least  Least Common Multiple
∴ we are looking for L.C.M of 18 and 24.
Step 1:
6 1 8
2
3
4
4
L.C.M of 18 and 24
=6X3X4
= 72
A bus and a minibus leave at the same time again after at least
72
minutes
11:00 a.m. + 72 minutes
= 11:72
= 12:12 p.m.
A bus and a minibus will leave at the same time again at
12:12
in the
afternoon
.