Gujarat Board Solutions Class 10 Maths Chapter 1 Real Numbers Ex 1.2
Gujarat Board Solutions Class 10 Maths Chapter 1 Real Numbers Ex 1.2
Gujarat Board Textbook Solutions Class 10 Maths Chapter 1 Real Numbers Ex 1.2
Question 1.
Find the LCM and HCF of the following integers by applying the prime factorisation method.
- 12, 15 and 21
- 17, 23 and 29
- 8, 9 and 25
Solution:
1. Prime factorisation of 12 = 2 × 2 × 3
Prime factorisation of 15 = 3 × 5
Prime factorisation of 21 = 3 × 7
HCF of 12, 15 and 21 = 3
LCM of 12, 15 and 21
= 2 × 2 × 3 × 5 × 7 = 420
2. Prime factorisation of 17 = 17 × 1
Prime factorisation of 23 = 23 × 1
Prime factorisation of 29 = 29 × 1
HCF of 12, 15, 29 = 1
LCM of 17, 23, 29 = 17 × 23 × 29
= 11339
3. Prime factorisation of 8 = 2 × 2 × 2
Prime factorisation of 9 = 3 × 3
Prime factorisation of 23 = 23 × 1
HCF of 8, 9 and 23 = 1
LCM of 8, 9, 23 = 2 × 2 × 2 × 3 × 3 × 23
= 1656
Question 2.
Given that HCF (306, 657) = 9, find LCM (306, 657).
Solution:
Given that HCF (306, 657) = 9
LCM × HCF = Product of two numbers
LCM × 9 = 306 × 657
LCM = 306×657/9
LCM = 34 × 657
LCM = 22338
∴ LCM of (306, 657) = 22338.
Question 3.
Check whether 6n can end with the digit 0 for any natural number n.(CBSE)
Solution:
6n will end with the digit zero if 6n is divisible by 2 and 5.
But 6n = (2 × 3)n = 2n × 3n
i.e. in the factorisation of 6n, no factor is of 5. Therefore by the fundamental theorem of arithmetic every composite number can be expressed a product of primes and this factorisation is unique apart from the order in which the prime factors occur.
Therefore our assumption is wrong that 6n ends in zero, thus there does not exist any natural number n for which 6n ends with zero.
Question 4.
Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.
Solution:
Given that 7 × 11 × 13 + 13
= (7 × 11 × 1 + 1) × 13
= (77 + 1) × 13
= 78 × 13
Composite number because it is product of more than two factors.
7 × 6 × 5 × 4 × 3 × 2 × 1 + 5
= (7 × 6 × 4 × 3 × 2 × 2 + 1) × 5
= (1008 + 1) × 5
= 1009 × 5
Product of more than two factors, which is a composite number.
Follow on Facebook page – Click Here
Google News join in – Click Here
Read More Asia News – Click Here
Read More Sports News – Click Here
Read More Crypto News – Click Here