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 Page 1


IMPORTANT POINTS 
Factor of a given number is that number by which the given number can be divided 
completely. 
1. Prime Numbers : 
A Natural number, which is divisible by 1 (one) and itself only is called a prime 
number. 
2. Highest Common Factor : 
H.C.F. stands for Highest Common Factor and H.C.F. of two or more given 
numbers is the greatest number (factor) which divides each given number 
completely. 
3. Lowest Common Factor : 
L.C.M. stands for Lowest Common Multiple. The L.C.M. of two or more given 
numbers is the lowest (smallest) number which is exactly divisible by each of the 
given numbers. 
EXERCISE 8(A) 
Question 1. 
Write all the factors of : 
(i) 15 
(ii) 55 
(iii) 48 
(iv) 36 
(v) 84 
Solution: 
(i) Factors of 15 = F 15 = 1, 3, 5 and 15 
(ii) Factors of 55 = F 55 = 1, 5, 11 and 55 
(iii) Factors of 48 = F 48 = 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48 
(iv) Factors of 36 = F 56 = 1, 2, 3, 4, 6, 9, 12, 18 and 36. 
(v) Factors of 84 = F 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 and 84. 
Question 2. 
Write all prime numbers : 
(i) less than 25 
(ii) between 15 and 35 
(iii) between 8 and 76 
Solution: 
(i) 2, 3, 5, 7, 11, 13, 17, 19 and 23 
(ii) 17, 19, 23, 29 and 31 
(iii) 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71 and 73. 
Page 2


IMPORTANT POINTS 
Factor of a given number is that number by which the given number can be divided 
completely. 
1. Prime Numbers : 
A Natural number, which is divisible by 1 (one) and itself only is called a prime 
number. 
2. Highest Common Factor : 
H.C.F. stands for Highest Common Factor and H.C.F. of two or more given 
numbers is the greatest number (factor) which divides each given number 
completely. 
3. Lowest Common Factor : 
L.C.M. stands for Lowest Common Multiple. The L.C.M. of two or more given 
numbers is the lowest (smallest) number which is exactly divisible by each of the 
given numbers. 
EXERCISE 8(A) 
Question 1. 
Write all the factors of : 
(i) 15 
(ii) 55 
(iii) 48 
(iv) 36 
(v) 84 
Solution: 
(i) Factors of 15 = F 15 = 1, 3, 5 and 15 
(ii) Factors of 55 = F 55 = 1, 5, 11 and 55 
(iii) Factors of 48 = F 48 = 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48 
(iv) Factors of 36 = F 56 = 1, 2, 3, 4, 6, 9, 12, 18 and 36. 
(v) Factors of 84 = F 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 and 84. 
Question 2. 
Write all prime numbers : 
(i) less than 25 
(ii) between 15 and 35 
(iii) between 8 and 76 
Solution: 
(i) 2, 3, 5, 7, 11, 13, 17, 19 and 23 
(ii) 17, 19, 23, 29 and 31 
(iii) 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71 and 73. 
Question 3. 
Write the prime-numbers from : 
(i) 5 to 45 
(ii) 2 to 32 
(iii) 8 to 48 
(iv) 9 to 59 
Solution: 
(i) 7, 11, 13, 17, 19, 23, 29, 31, 37, 41 and 43. 
(ii) 3, 5, 7, 11, 13, 17, 19, 23 29 and 31. 
(iii) 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47. 
(iv) 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 and 53. 
Question 4. 
Write the prime factors of: 
(i) 16 
(ii) 27 
(iii) 35 
(iv) 49 
Solution: 
(i) Prime factors of 16 = 2 
 
(ii) Prime factors of 27 = 3 
 
(iii) Prime factors of 35 = 5, 7 
 
(iv) Prime factors of 49 = 7 
 
Page 3


IMPORTANT POINTS 
Factor of a given number is that number by which the given number can be divided 
completely. 
1. Prime Numbers : 
A Natural number, which is divisible by 1 (one) and itself only is called a prime 
number. 
2. Highest Common Factor : 
H.C.F. stands for Highest Common Factor and H.C.F. of two or more given 
numbers is the greatest number (factor) which divides each given number 
completely. 
3. Lowest Common Factor : 
L.C.M. stands for Lowest Common Multiple. The L.C.M. of two or more given 
numbers is the lowest (smallest) number which is exactly divisible by each of the 
given numbers. 
EXERCISE 8(A) 
Question 1. 
Write all the factors of : 
(i) 15 
(ii) 55 
(iii) 48 
(iv) 36 
(v) 84 
Solution: 
(i) Factors of 15 = F 15 = 1, 3, 5 and 15 
(ii) Factors of 55 = F 55 = 1, 5, 11 and 55 
(iii) Factors of 48 = F 48 = 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48 
(iv) Factors of 36 = F 56 = 1, 2, 3, 4, 6, 9, 12, 18 and 36. 
(v) Factors of 84 = F 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 and 84. 
Question 2. 
Write all prime numbers : 
(i) less than 25 
(ii) between 15 and 35 
(iii) between 8 and 76 
Solution: 
(i) 2, 3, 5, 7, 11, 13, 17, 19 and 23 
(ii) 17, 19, 23, 29 and 31 
(iii) 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71 and 73. 
Question 3. 
Write the prime-numbers from : 
(i) 5 to 45 
(ii) 2 to 32 
(iii) 8 to 48 
(iv) 9 to 59 
Solution: 
(i) 7, 11, 13, 17, 19, 23, 29, 31, 37, 41 and 43. 
(ii) 3, 5, 7, 11, 13, 17, 19, 23 29 and 31. 
(iii) 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47. 
(iv) 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 and 53. 
Question 4. 
Write the prime factors of: 
(i) 16 
(ii) 27 
(iii) 35 
(iv) 49 
Solution: 
(i) Prime factors of 16 = 2 
 
(ii) Prime factors of 27 = 3 
 
(iii) Prime factors of 35 = 5, 7 
 
(iv) Prime factors of 49 = 7 
 
Question 5. 
If P n means prime factors of n, find: 
(i) p 6 
(ii) P 24 
(iii) p 50 
(iv) P 42 
Solution: 
(i) F 6 = 1, 2, 3, 6 
P.F 6 (Prime factor of 6) = 2 and 3. 
(ii) F 24 = 1, 2, 3, 4, 6, 8, 12, 24 
P.F 24 = 2 and 3. 
(iii) F 50 = 2, 5, 5 
P.F 50 = 2 and 5. 
(iv) F 42 = 1, 2, 3, 6, 7, 14, 21, 42 
P.F 42 = 2, 3 and 7. 
EXERCISE 8(B) 
Question 1. 
Using the common factor method, find the H.C.F. of : 
(i) 16 and 35 
(ii) 25 and 20 
(iii) 27 and 75 
(iv) 8, 12 and 18 
(v) 24, 36, 45 and 60 
Solution: 
(i) F 16 = 1, 2, 4, 8, 16 
F 35 = 1, 5, 7, 35 
Common factors between 16 and 35 = 1 
H.C.F. of 16 and 35 = 1 
(ii) F 25 = 1, 5, 25 
F 20 = 1, 2, 4, 5, 10, 20 
Common factors between 25 and 20 = 1, 5 
H.C.F. of 25 and 20 = 5 
(iii) F 27 = 1, 3, 9, 27 
F 75 = 1, 3, 5, 15, 25, 75 
Common factors between 27 and 75 = 1, 3 
H.C.F. of 27 and 75 = 3 
(iv) F 8 = 1, 2, 4, 8 
F 12 = 1, 2, 3, 4, 6, 12 
F 18 = 1, 2, 3, 6, 9, 18 
Common factors between 8, 12 and 18 = 1, 2 
H.C.F. of 8, 12 and 18 = 2 
(v) F 24 = 1, 2, 3, 4, 6, 8, 12, 24 
F 36 = 1, 2, 3, 4, 6, 12, 18, 36 
F 45 = 1, 3, 5, 9, 15, 45 
Page 4


IMPORTANT POINTS 
Factor of a given number is that number by which the given number can be divided 
completely. 
1. Prime Numbers : 
A Natural number, which is divisible by 1 (one) and itself only is called a prime 
number. 
2. Highest Common Factor : 
H.C.F. stands for Highest Common Factor and H.C.F. of two or more given 
numbers is the greatest number (factor) which divides each given number 
completely. 
3. Lowest Common Factor : 
L.C.M. stands for Lowest Common Multiple. The L.C.M. of two or more given 
numbers is the lowest (smallest) number which is exactly divisible by each of the 
given numbers. 
EXERCISE 8(A) 
Question 1. 
Write all the factors of : 
(i) 15 
(ii) 55 
(iii) 48 
(iv) 36 
(v) 84 
Solution: 
(i) Factors of 15 = F 15 = 1, 3, 5 and 15 
(ii) Factors of 55 = F 55 = 1, 5, 11 and 55 
(iii) Factors of 48 = F 48 = 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48 
(iv) Factors of 36 = F 56 = 1, 2, 3, 4, 6, 9, 12, 18 and 36. 
(v) Factors of 84 = F 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 and 84. 
Question 2. 
Write all prime numbers : 
(i) less than 25 
(ii) between 15 and 35 
(iii) between 8 and 76 
Solution: 
(i) 2, 3, 5, 7, 11, 13, 17, 19 and 23 
(ii) 17, 19, 23, 29 and 31 
(iii) 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71 and 73. 
Question 3. 
Write the prime-numbers from : 
(i) 5 to 45 
(ii) 2 to 32 
(iii) 8 to 48 
(iv) 9 to 59 
Solution: 
(i) 7, 11, 13, 17, 19, 23, 29, 31, 37, 41 and 43. 
(ii) 3, 5, 7, 11, 13, 17, 19, 23 29 and 31. 
(iii) 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47. 
(iv) 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 and 53. 
Question 4. 
Write the prime factors of: 
(i) 16 
(ii) 27 
(iii) 35 
(iv) 49 
Solution: 
(i) Prime factors of 16 = 2 
 
(ii) Prime factors of 27 = 3 
 
(iii) Prime factors of 35 = 5, 7 
 
(iv) Prime factors of 49 = 7 
 
Question 5. 
If P n means prime factors of n, find: 
(i) p 6 
(ii) P 24 
(iii) p 50 
(iv) P 42 
Solution: 
(i) F 6 = 1, 2, 3, 6 
P.F 6 (Prime factor of 6) = 2 and 3. 
(ii) F 24 = 1, 2, 3, 4, 6, 8, 12, 24 
P.F 24 = 2 and 3. 
(iii) F 50 = 2, 5, 5 
P.F 50 = 2 and 5. 
(iv) F 42 = 1, 2, 3, 6, 7, 14, 21, 42 
P.F 42 = 2, 3 and 7. 
EXERCISE 8(B) 
Question 1. 
Using the common factor method, find the H.C.F. of : 
(i) 16 and 35 
(ii) 25 and 20 
(iii) 27 and 75 
(iv) 8, 12 and 18 
(v) 24, 36, 45 and 60 
Solution: 
(i) F 16 = 1, 2, 4, 8, 16 
F 35 = 1, 5, 7, 35 
Common factors between 16 and 35 = 1 
H.C.F. of 16 and 35 = 1 
(ii) F 25 = 1, 5, 25 
F 20 = 1, 2, 4, 5, 10, 20 
Common factors between 25 and 20 = 1, 5 
H.C.F. of 25 and 20 = 5 
(iii) F 27 = 1, 3, 9, 27 
F 75 = 1, 3, 5, 15, 25, 75 
Common factors between 27 and 75 = 1, 3 
H.C.F. of 27 and 75 = 3 
(iv) F 8 = 1, 2, 4, 8 
F 12 = 1, 2, 3, 4, 6, 12 
F 18 = 1, 2, 3, 6, 9, 18 
Common factors between 8, 12 and 18 = 1, 2 
H.C.F. of 8, 12 and 18 = 2 
(v) F 24 = 1, 2, 3, 4, 6, 8, 12, 24 
F 36 = 1, 2, 3, 4, 6, 12, 18, 36 
F 45 = 1, 3, 5, 9, 15, 45 
F 60 = 1, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 
Common factor between 24, 36, 45 and 60 = 1, 3 
H.C.F. of 24, 36, 45 and 60 = 3 
Question 2. 
Using the prime factor method, find the H.C.F. of: 
(i) 5 and 8 
(ii) 24 and 49 
(iii) 40, 60 and 80 
(iv) 48, 84 and 88 
(v) 12, 16 and 28 
Solution: 
(i) Prime factor of 5 = 5 
Prime factor of 8 = 2 x 2 x 2 
No common prime factor 
H.C.F. of 5 and 8 = 1 
(as both the number are co-prime) 
(ii) Prime factor of 24 = 2 x 2 x 2 x 3 
Prime factor of 49 = 7 x 7 
No common prime factor, number are co-prime. 
H.C.F. of 24 and 49 = 1. 
(iii) Prime factor of 40 = 2 x 2 x 2 x 5 
Prime factor of 60 = 2 x 2 x 3 x 5 
Prime factor of 80 = 2 x 2 x 2 x 2 x 5 
Common prime factor = 2 x 2 x 5 
H.C.F. of 40, 60 and 80 = 2 x 2 x 5 = 20 
(iv) Prime factor of 48 = 2 x 2 x 2 x 2 x 3 
Prime factor of 84 = 2 x 2 x 3 x 7 
Prime factor of 88 = 2 x 2 x 2 x 11 
Common prime factor of 48, 84 and 88 = 2 x 2 
H.C.F. of 48, 84 and 88 = 2 x 2 = 4 
(v) Prime factor of 12 = 2 x 2 x 3 
Prime factor of 16 = 2 x 2 x 2 x 2 
Prime factor of 28 = 2 x 2 x 7 
Common prime factor between 12, 16 and 28 = 2 x 2 
H.C.F. of 12, 16 and 28 = 2 x 2 = 4 
Question 3. 
Using the division method, find the H.C.F. of the following : 
(i) 16 and 24 
(ii) 18 and 30 
(iii) 7, 14 and 24 
(iv) 70,80,120 and 150 
(v) 32, 56 and 46 
Solution: 
Page 5


IMPORTANT POINTS 
Factor of a given number is that number by which the given number can be divided 
completely. 
1. Prime Numbers : 
A Natural number, which is divisible by 1 (one) and itself only is called a prime 
number. 
2. Highest Common Factor : 
H.C.F. stands for Highest Common Factor and H.C.F. of two or more given 
numbers is the greatest number (factor) which divides each given number 
completely. 
3. Lowest Common Factor : 
L.C.M. stands for Lowest Common Multiple. The L.C.M. of two or more given 
numbers is the lowest (smallest) number which is exactly divisible by each of the 
given numbers. 
EXERCISE 8(A) 
Question 1. 
Write all the factors of : 
(i) 15 
(ii) 55 
(iii) 48 
(iv) 36 
(v) 84 
Solution: 
(i) Factors of 15 = F 15 = 1, 3, 5 and 15 
(ii) Factors of 55 = F 55 = 1, 5, 11 and 55 
(iii) Factors of 48 = F 48 = 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48 
(iv) Factors of 36 = F 56 = 1, 2, 3, 4, 6, 9, 12, 18 and 36. 
(v) Factors of 84 = F 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 and 84. 
Question 2. 
Write all prime numbers : 
(i) less than 25 
(ii) between 15 and 35 
(iii) between 8 and 76 
Solution: 
(i) 2, 3, 5, 7, 11, 13, 17, 19 and 23 
(ii) 17, 19, 23, 29 and 31 
(iii) 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71 and 73. 
Question 3. 
Write the prime-numbers from : 
(i) 5 to 45 
(ii) 2 to 32 
(iii) 8 to 48 
(iv) 9 to 59 
Solution: 
(i) 7, 11, 13, 17, 19, 23, 29, 31, 37, 41 and 43. 
(ii) 3, 5, 7, 11, 13, 17, 19, 23 29 and 31. 
(iii) 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47. 
(iv) 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 and 53. 
Question 4. 
Write the prime factors of: 
(i) 16 
(ii) 27 
(iii) 35 
(iv) 49 
Solution: 
(i) Prime factors of 16 = 2 
 
(ii) Prime factors of 27 = 3 
 
(iii) Prime factors of 35 = 5, 7 
 
(iv) Prime factors of 49 = 7 
 
Question 5. 
If P n means prime factors of n, find: 
(i) p 6 
(ii) P 24 
(iii) p 50 
(iv) P 42 
Solution: 
(i) F 6 = 1, 2, 3, 6 
P.F 6 (Prime factor of 6) = 2 and 3. 
(ii) F 24 = 1, 2, 3, 4, 6, 8, 12, 24 
P.F 24 = 2 and 3. 
(iii) F 50 = 2, 5, 5 
P.F 50 = 2 and 5. 
(iv) F 42 = 1, 2, 3, 6, 7, 14, 21, 42 
P.F 42 = 2, 3 and 7. 
EXERCISE 8(B) 
Question 1. 
Using the common factor method, find the H.C.F. of : 
(i) 16 and 35 
(ii) 25 and 20 
(iii) 27 and 75 
(iv) 8, 12 and 18 
(v) 24, 36, 45 and 60 
Solution: 
(i) F 16 = 1, 2, 4, 8, 16 
F 35 = 1, 5, 7, 35 
Common factors between 16 and 35 = 1 
H.C.F. of 16 and 35 = 1 
(ii) F 25 = 1, 5, 25 
F 20 = 1, 2, 4, 5, 10, 20 
Common factors between 25 and 20 = 1, 5 
H.C.F. of 25 and 20 = 5 
(iii) F 27 = 1, 3, 9, 27 
F 75 = 1, 3, 5, 15, 25, 75 
Common factors between 27 and 75 = 1, 3 
H.C.F. of 27 and 75 = 3 
(iv) F 8 = 1, 2, 4, 8 
F 12 = 1, 2, 3, 4, 6, 12 
F 18 = 1, 2, 3, 6, 9, 18 
Common factors between 8, 12 and 18 = 1, 2 
H.C.F. of 8, 12 and 18 = 2 
(v) F 24 = 1, 2, 3, 4, 6, 8, 12, 24 
F 36 = 1, 2, 3, 4, 6, 12, 18, 36 
F 45 = 1, 3, 5, 9, 15, 45 
F 60 = 1, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 
Common factor between 24, 36, 45 and 60 = 1, 3 
H.C.F. of 24, 36, 45 and 60 = 3 
Question 2. 
Using the prime factor method, find the H.C.F. of: 
(i) 5 and 8 
(ii) 24 and 49 
(iii) 40, 60 and 80 
(iv) 48, 84 and 88 
(v) 12, 16 and 28 
Solution: 
(i) Prime factor of 5 = 5 
Prime factor of 8 = 2 x 2 x 2 
No common prime factor 
H.C.F. of 5 and 8 = 1 
(as both the number are co-prime) 
(ii) Prime factor of 24 = 2 x 2 x 2 x 3 
Prime factor of 49 = 7 x 7 
No common prime factor, number are co-prime. 
H.C.F. of 24 and 49 = 1. 
(iii) Prime factor of 40 = 2 x 2 x 2 x 5 
Prime factor of 60 = 2 x 2 x 3 x 5 
Prime factor of 80 = 2 x 2 x 2 x 2 x 5 
Common prime factor = 2 x 2 x 5 
H.C.F. of 40, 60 and 80 = 2 x 2 x 5 = 20 
(iv) Prime factor of 48 = 2 x 2 x 2 x 2 x 3 
Prime factor of 84 = 2 x 2 x 3 x 7 
Prime factor of 88 = 2 x 2 x 2 x 11 
Common prime factor of 48, 84 and 88 = 2 x 2 
H.C.F. of 48, 84 and 88 = 2 x 2 = 4 
(v) Prime factor of 12 = 2 x 2 x 3 
Prime factor of 16 = 2 x 2 x 2 x 2 
Prime factor of 28 = 2 x 2 x 7 
Common prime factor between 12, 16 and 28 = 2 x 2 
H.C.F. of 12, 16 and 28 = 2 x 2 = 4 
Question 3. 
Using the division method, find the H.C.F. of the following : 
(i) 16 and 24 
(ii) 18 and 30 
(iii) 7, 14 and 24 
(iv) 70,80,120 and 150 
(v) 32, 56 and 46 
Solution:

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