CHAPTER – 1
REAL NUMBERS
The Fundamental Theorem of Arithmetic
Every composite number can be expressed ( factorised) as a product of primes, and this factorisation
is unique, apart from the order in which the prime factors occur.
The prime factorisation of a natural number is unique, except for the order of its factors.
Property of HCF and LCM of two positive integers ‘a’ and ‘b’:
HCF (a, b) LCM (a, b) a b
ab
LCM (a, b)
HCF (a, b)
a b
HCF (a, b)
LCM (a, b)
PRIME FACTORISATION METHOD TO FIND HCF AND LCM
HCF(a, b) = Product of the smallest power of each common prime factor in the numbers.
LCM(a, b) = Product of the greatest power of each prime factor, involved in the numbers.
IMPORTANT QUESTIONS
Find the LCM and HCF of 510 and 92 and verify that LCM × HCF = product of the two
numbers
Solution: 510 = 2 x 3 x 5 x 17
92 = 2 x 2 x 23 = 22 x 23
HCF = 2
LCM = 22 x 3 x 5 x 17 x 23 = 23460
Product of two numbers = 510 x 92 = 46920
HCF x LCM = 2 x 23460 = 46920
Hence, product of two numbers = HCF × LCM
Questions for practice
1. Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.
2. Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM.
3. Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product
of the two numbers: (i) 26 and 91 (ii) 336 and 54
4. Find the LCM and HCF of the following integers by applying the prime factorisation method: (i)
12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25
5. Explain why 3 × 5 × 7 + 7 is a composite number.
6. Can the number 6n, n being a natural number, end with the digit 5? Give reasons.
7. Can the number 4n, n being a natural number, end with the digit 0? Give reasons.
8. Given that HCF (306, 657) = 9, find LCM (306, 657).
9. If two positive integers a and b are written as a = x3y2 and b = xy3; x, y are prime numbers, then
find the HCF (a, b).
10. If two positive integers p and q can be expressed as p = ab2 and q = a3b; a, b being prime
numbers, then find the LCM (p, q).
11. Find the largest number which divides 245 and 1029 leaving remainder 5 in each case.
12. Find the largest number which divides 2053 and 967 and leaves a remainder of 5 and 7
respectively.
REAL NUMBERS
The Fundamental Theorem of Arithmetic
Every composite number can be expressed ( factorised) as a product of primes, and this factorisation
is unique, apart from the order in which the prime factors occur.
The prime factorisation of a natural number is unique, except for the order of its factors.
Property of HCF and LCM of two positive integers ‘a’ and ‘b’:
HCF (a, b) LCM (a, b) a b
ab
LCM (a, b)
HCF (a, b)
a b
HCF (a, b)
LCM (a, b)
PRIME FACTORISATION METHOD TO FIND HCF AND LCM
HCF(a, b) = Product of the smallest power of each common prime factor in the numbers.
LCM(a, b) = Product of the greatest power of each prime factor, involved in the numbers.
IMPORTANT QUESTIONS
Find the LCM and HCF of 510 and 92 and verify that LCM × HCF = product of the two
numbers
Solution: 510 = 2 x 3 x 5 x 17
92 = 2 x 2 x 23 = 22 x 23
HCF = 2
LCM = 22 x 3 x 5 x 17 x 23 = 23460
Product of two numbers = 510 x 92 = 46920
HCF x LCM = 2 x 23460 = 46920
Hence, product of two numbers = HCF × LCM
Questions for practice
1. Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.
2. Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM.
3. Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product
of the two numbers: (i) 26 and 91 (ii) 336 and 54
4. Find the LCM and HCF of the following integers by applying the prime factorisation method: (i)
12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25
5. Explain why 3 × 5 × 7 + 7 is a composite number.
6. Can the number 6n, n being a natural number, end with the digit 5? Give reasons.
7. Can the number 4n, n being a natural number, end with the digit 0? Give reasons.
8. Given that HCF (306, 657) = 9, find LCM (306, 657).
9. If two positive integers a and b are written as a = x3y2 and b = xy3; x, y are prime numbers, then
find the HCF (a, b).
10. If two positive integers p and q can be expressed as p = ab2 and q = a3b; a, b being prime
numbers, then find the LCM (p, q).
11. Find the largest number which divides 245 and 1029 leaving remainder 5 in each case.
12. Find the largest number which divides 2053 and 967 and leaves a remainder of 5 and 7
respectively.
