Class 10 Maths NCERT Chapter 1. Real Numbers Solutions

Class 10 Maths NCERT Complete Solutions Chapter wise

Real Numbers Ex 1.1

Real Numbers Ex 1.2

NCERT Class 10 Maths Ch.1 Real Numbers Ex 1.1

Question 1. Express each number as a product of its prime factors: (i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429

Solution: (i) 140

To find the prime factors of 140, we perform prime factorization:

the prime factorization of 140 is 2² × 5 × 7.

(ii) 156

Prime factorization of 156:

The prime factorization of 156 is 2²×3×13.

(iii) 3825

Prime factorization of 3825:

The prime factorization of 3825 is 3²×5²×17.

(v) 7429

Prime factorization of 7429:

The prime factorization of 7429 is 17×19×23.

Question 2. 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) 510 and 92 (iii) 336 and 54

Solution: (i) 26 and 91

Prime factorization of 26: 2×13

Prime factorization of 91: 7×13

HCF (Highest Common Factor) is the product of the smallest power of each common prime factor:

HCF(26, 91) = 13¹=13

LCM (Least Common Multiple) is the product of the greatest power of each prime factor involved:

LCM(26, 91) = 2¹×7¹×13¹=182

Verification: Product of the numbers = 26×91=2366

Product of LCM and HCF = 182×13=2366

Since 2366=2366, the verification holds true.

(ii) 510 and 92

Prime factorization of 510: 2×3×5×17

Prime factorization of 92: 2²×23

HCF(510, 92) = 2¹=2

LCM(510, 92) = 2²×3×5×17×23=23460

Verification: Product of the numbers = 510×92=46920

Product of LCM and HCF = 23460×2=46920 Since 46920=46920, the verification holds true.

(iii) 336 and 54

Prime factorization of 336: 2⁴×3×7

Prime factorization of 54: 2×3³

HCF(336, 54) = 2¹×3¹=6

LCM(336, 54) = 2⁴×3³×7=16×27×7=3024

Verification: Product of the numbers = 336×54=18144

Product of LCM and HCF = 3024×6=18144

Since 18144=18144, the verification holds true.

Question 3. Find the LCM and HCF of the following integers by applying the prime factorization method. (i) 12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25

Q4. Given that HCF(306,657)=9, find LCM(306,657).

Answer:

Q5. Check whether 6n can end with the digit 0 for any natural number n.

Solution:

Q6. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.

Solution:

Q7. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?

Solution: