Lowest Common Multiple
& Highest Common Factor
Digits:
The figures which make up a number are called digits.
Q.1. What is factor?
Ans: The numbers by which a larger number can be divided are the factors of that number.
Q.2. What is G.C.D?
Ans: The greatest divisor that lies among the common factors of two numbers is the G.C.D.
Q. 3. What is multiple?
Ans: When a number is divisible by another number, the first number is called the multiple of the other number.
For example, in the number 627, 6 is called the hundred digit, 2 is called the ten-digit, and 7 the unit digit.
A multiple of a number is exactly divisible by that number.
For example, 24 is a multiple of 4. If a number is multiple of a second number, the second number is said to be a factor of the first.
A prime number is a number which has no factors other than itself and unity.
The first five prime numbers are 2, 3, 5, 7, 11.
A prime factor is a factor which is a prime number.
The prime factors of 24 are 2, 2, 2, and 3.
Expressed in its prime factors ,
24 = 2 x 2 x 2 x 3
Ex. 7.4
1. List 3 multiples for the following numbers, ordering them from the smallest.
a) 4
Solution: 4 x 1 = 4
4 x 2 = 8
4 x 3 = 12
2. List 3 common multiples for the following pairs of numbers, ordering them from the smallest. also write the least common multiples (LCM).
a) 3, 4
Solution: Multiples of 3 : 3, 4, 6, 9, 12, 15, 18, 24, 27, 30, 33, 36,............
Multiples of 4 : 4, 8, 12, 16, 20, 24, 28, 32, 36,...........
3 common multiples of 3 and 4 are 12, 24, 36
Ordering from smallest to highest 12, 24, 36
The smallest common multiple is 12
The LCM of 3 and 4 = 12 Ans.
4. List all the common factors of the following numbers. also write the highest common factors (HCF) for each pair.
a) 9, 15
Solution: 9 = 1 x 9 = 3 x 3
The factors of 9 : 1, 3, 9
15 = 1 x 15
= 3 x 5
The factors of 15 : 1 ,3, 5, 15
The common factors of 9 and 15 are 1 and 3.
The highest common factors (HCF) for 9 and 15 is 3. Ans.
5. Find the least common multiple (LCM) and highest common factors (HCF) for each .
a) 8, 12, 24
Solution: 8 = 4 x 2
= 2 x 2 x 2
12 = 2 x 6
= 2 x 2 x 3
24 = 2 x 12
= 2 x 4 x 3
= 2 x 2 x 2 x 3
LCM = 2 x 2 x 2 x 3 = 24
Again, 8 = 1 x 8
= 2 x 4
and 12 = 1 x 12
= 2 x 6
= 3 x 4
and 24 = 1 x 24
= 2 x 12
= 3 x 8
= 4 x 6
HCF = 4
Highest Common Factor:
The Highest Common Factor (H.C.F) of two or more numbers is the greatest number which is a factor of each of them.
Ex. Find the H.C.F and L.C.M. of 63, 441, 3969.
Solution: 63 = 3 x 21
= 3 x 3 x 7 441
= 3 x 147
= 3 x 3 x 49
= 3 x 3 x 7 x 7 3969
= 3 x 1323
= 3 x 3 x 441
= 3 x 3 x 3 x 3 x 7 x 7
H.C.F. = 3 x 3 x 7 = 63
L.C.M. = 3 x 3 x 3 x 3 x 7 x 7
= 3969
Lowest Common Multiple:
The Lowest Common Multiple (L.C.M.) of two or more numbers is the least number which is a multiple of each of them.