The document discusses concepts related to factors, multiples, highest common factors (HCF), and least common multiples (LCM) of numbers. It provides definitions and examples of factors, multiples, HCF, and LCM. Several examples of finding the HCF and LCM of numbers using factorization and other methods are shown. The document also presents solutions to example problems involving HCF, LCM, ratios, remainders, and word problems related to these concepts.

1. Dipto Shaha
BBA,MajorinMarketing

2. Least Common Multiples (L.C.M)
Highest Common Factors (H.C.F)
Factors Multiples
If a number or group of
numbers exactly divide a
particular given number
are called factors.
If a particular number is
multiplied with 1,2,3, or
any other numbers then
the products are called
multiples of that given
number.1,2,3,4,6,8,12,and 24 are the
factors of 24.
24,48,72 and 96 are the
multiples of 24.

3. Highest Common Factors (H.C.F)
The greatest of the factors among the common factors of the given
numbers is called Highest Common Factor of the given
numbers.
Find the H.C.F of 48 and 72
The factors of 48 are 1 , 2 , 3 , 4 , 6 , 8 , 12 , 16 , 24 , 48 .
The factors of 72 are 1 , 2 , 3 , 4 , 6 , 8 , 9 , 12 , 18 , 24 , 36,
72 .
So the highest common factor is 24

4. The ways of finding H.C.F
Factorization Method
Division Method
Find the H.C.F of (using both methods )
 42, 63 and 140
 108,288 and 360
Express each one of the given numbers
into product of the prime factors .
Then H.C.F is the product of least
powers of common prime factors.

5. Least Common Multiples (L.C.M)
The lowest number which is exactly divisible by each one of the
given numbers is called Least Common Multiples of the given
numbers.
Find the L.C.M of 48 and 72
So the least common multiple is 144
The multiples of 48 are 48 , 96 ,144 ,192 , 240, 288 , 336,384, 432.
The multiples of 72 are, 72, 144 , 216, 288, 360 , 432

6. The ways of finding L.C.M
Factorization Method
Common Division Method
Find the L.C.M of (using both methods )
 16,24,36 and 54
Resolve each one of the given numbers
into product of the prime factors .
Then L.C.M is the product of highest
powers of all the factors.

7. Lets solve some problems in H.C.F & L.C.M
 Find the H.C.F & L.C.M 2
3
,
8
9
,
16
81
, 𝑎𝑛𝑑
10
27
H.C.F of the given function =
𝐻.𝐶.𝐹 𝑜𝑓 2,8,16,10
𝐿.𝐶.𝑀 𝑜𝑓 3,9,81,27
=
2
81
L.C.M of the given function =
𝐿.𝐶.𝑀 𝑜𝑓2,8,16,10
𝐻.𝐶.𝐹 𝑜𝑓 3,9,81,27
=
80
13
 Reduce
391
667
to lowest terms ?
SOL:
𝟏𝟕
𝟐𝟗

8. Lets solve some problems in H.C.F & L.C.M ( Continue…)
 Tow numbers are in the ratio of 15: 11 . If their H.C.F is 13 , find
the numbers.
 Tow numbers are in the ratio of 3 : 4 . If their L.C.M is 84 , find
the numbers.
SOL: The numbers are 195 and 143
SOL: The numbers are 21 and 28
 How many factors of 560 ?
SOL: The number of factor of 560 is 20.

9. Lets solve some problems in H.C.F & L.C.M ( Continue…)
 If the sum of the two numbers is 462 and their H.C.F is 22 . What
is the minimum number of the pairs that satisfy these conditions ?
SOL: The number of such pair is 6.
 If the sum and difference of the L.C.M and H.C.F of two
numbers are 592 and 518 respectively. It the sum of the numbers
be 296 , find the numbers?
SOL: The numbers are 111 and 185 .
 The H.C.F of two numbers is 11 and L.C.M is 639 . If one of the
numbers is 77, find the other.
SOL: The other number is 99

10. Lets solve some problems in H.C.F & L.C.M ( Continue…)
 Find the greatest number which on dividing 1657 and 2037 leaves
remainders 6 and 5 respectively.
SOL: Required number is 127.
 Find the greatest number which divides 62, 132, 237 to leave the same
remainder in each case?
SOL: Required number is 35.
 Find the least number which divided by 6, 7, 8, 9 , 12 leaves the same 1
remainder in each case.
SOL: Required number is (504+1)= 505.
 Find the smallest number which when increased by 10 is completely
divisible by 12, 15, 18, 20 and 24.
SOL: Required number is 350.

11. Lets solve some problems in H.C.F & L.C.M ( Continue…)
 Find the greatest number of five digits which is divisible by 15, 21, 36.
SOL: Required number is ( 99999 – 459 ) =99540.
 Find the smallest number of five digits exactly divisible by 16, 24, 36
and 54.
SOL: Required number is 10000 +(432 - 64 ) = 10368.
 Find the largest number which , when subtracted from 10000 , the
remainder is divisible by 32, 36, 48 and 54.
SOL: Required number is (10000 - 864 ) = 9136.
 what is the least number which when divided by the numbers 3, 5, 6, 8,
10 and 12 leaves in each case a remainder 2 but when divided by 13
leaves no remainder?
SOL: Required number is (120 * 8 + 2 ) = 962.

12. Lets solve some problems in H.C.F & L.C.M ( Continue…)
 A, B and C start at the same time at the same direction to run around a
circle stadium. A completes a round in 252 seconds, B in 308 seconds and
C in 198 seconds, all starting at the same point. After what time will they
meet again at the starting point?
SOL: 46 minutes 12 seconds.
 Arrange the fractions in the ascending order
17
18
43
45
31
36
59
60
SOL: 𝟑𝟏
𝟑𝟔
<
𝟏𝟕
𝟏𝟖
<
𝟒𝟑
𝟒𝟓
<
𝟓𝟗
𝟔𝟎

13. Lets solve some problems in H.C.F & L.C.M ( Continue…)
 A pendulum strikes 5 times in 3 seconds and another pendulum strikes
7 times at 5 seconds. If both pendulums start striking at the same time,
how many clear strikes can be listened in 1 minute?
SOL: Required number is (100 + 105) - 6= 199.
 The circumference of front wheel of a carriage is 4m and that of the
back wheel is 5m. The front wheel revolve 200 times more than the
back wheel to describe a distance of …..?
SOL: 4000 meter or 4 kilometer.