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NCERT Class 10 Mathematics Exercise 1.1 Solutions: Real Numbers

AI-written, checked on 23 Oct 2026

Solutions written by Shishya, checked — not the board's official answers

Step-by-step working for every question of Exercise 1.1 in NCERT's Class 10 Mathematics book, chapter “Real Numbers”, written by Shishya. Each question is in our own short words; open the NCERT chapter for the exact wording and figures.

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Ex 1.1 Q1
Write each number as a product of its prime factors: (i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429.
1. (i) 140 ÷ 2 = 70, 70 ÷ 2 = 35, 35 ÷ 5 = 7, 7 ÷ 7 = 1
  why?
  Every whole number greater than 1 is a product of primes in exactly one way (Fundamental Theorem of Arithmetic). Dividing again and again by a prime that goes in exactly finds that product.
2. (i) Prime factors are 2, 2, 5, 7
  why?
  Every whole number greater than 1 is a product of primes in exactly one way (Fundamental Theorem of Arithmetic). Dividing again and again by a prime that goes in exactly finds that product.
3. (ii) 156 ÷ 2 = 78, 78 ÷ 2 = 39, 39 ÷ 3 = 13, 13 ÷ 13 = 1
  why?
  Every whole number greater than 1 is a product of primes in exactly one way (Fundamental Theorem of Arithmetic). Dividing again and again by a prime that goes in exactly finds that product.
4. (ii) Prime factors are 2, 2, 3, 13
  why?
  Every whole number greater than 1 is a product of primes in exactly one way (Fundamental Theorem of Arithmetic). Dividing again and again by a prime that goes in exactly finds that product.
5. (iii) 3825 ÷ 3 = 1275, 1275 ÷ 3 = 425, 425 ÷ 5 = 85, 85 ÷ 5 = 17, 17 ÷ 17 = 1
  why?
  Every whole number greater than 1 is a product of primes in exactly one way (Fundamental Theorem of Arithmetic). Dividing again and again by a prime that goes in exactly finds that product.
6. (iii) Prime factors are 3, 3, 5, 5, 17
  why?
  Every whole number greater than 1 is a product of primes in exactly one way (Fundamental Theorem of Arithmetic). Dividing again and again by a prime that goes in exactly finds that product.
7. (iv) 5005 ÷ 5 = 1001, 1001 ÷ 7 = 143, 143 ÷ 11 = 13, 13 ÷ 13 = 1
  why?
  Every whole number greater than 1 is a product of primes in exactly one way (Fundamental Theorem of Arithmetic). Dividing again and again by a prime that goes in exactly finds that product.
8. (iv) Prime factors are 5, 7, 11, 13
  why?
  Every whole number greater than 1 is a product of primes in exactly one way (Fundamental Theorem of Arithmetic). Dividing again and again by a prime that goes in exactly finds that product.
9. (v) 7429 ÷ 17 = 437, 437 ÷ 19 = 23, 23 ÷ 23 = 1
  why?
  Every whole number greater than 1 is a product of primes in exactly one way (Fundamental Theorem of Arithmetic). Dividing again and again by a prime that goes in exactly finds that product.
10. (v) Prime factors are 17, 19, 23
  why?
  Every whole number greater than 1 is a product of primes in exactly one way (Fundamental Theorem of Arithmetic). Dividing again and again by a prime that goes in exactly finds that product.
Answer: (i) 2² × 5 × 7; (ii) 2² × 3 × 13; (iii) 3² × 5² × 17; (iv) 5 × 7 × 11 × 13; (v) 17 × 19 × 23
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Ex 1.1 Q2
Find the LCM and HCF of each pair and check that LCM × HCF = product of the two numbers: (i) 26 and 91 (ii) 510 and 92 (iii) 336 and 54.
1. (i) Prime factorization of 26 is 2 × 13 and of 91 is 7 × 13.
  why?
  Every whole number greater than 1 is a product of primes in exactly one way (Fundamental Theorem of Arithmetic). Dividing again and again by a prime that goes in exactly finds that product.
2. (i) HCF is the product of common factors, so HCF = 13.
  why?
  HCF = the product of the prime factors common to all the numbers, each taken with its smallest power.
3. (i) LCM is the product of the highest powers of all factors, so LCM = 2 × 7 × 13 = 182.
  why?
  LCM = the product of every prime factor that appears in any of the numbers, each taken with its greatest power.
4. (i) Check: LCM × HCF = 182 × 13 = 2366 and product of numbers = 26 × 91 = 2366.
  why?
  For any two positive whole numbers, HCF × LCM = the product of the two numbers, so this multiplication checks the HCF and LCM found.
5. (ii) Prime factorization of 510 is 2 × 3 × 5 × 17 and of 92 is 2 × 2 × 23.
  why?
  Every whole number greater than 1 is a product of primes in exactly one way (Fundamental Theorem of Arithmetic). Dividing again and again by a prime that goes in exactly finds that product.
6. (ii) HCF is the product of common factors, so HCF = 2.
  why?
  HCF = the product of the prime factors common to all the numbers, each taken with its smallest power.
7. (ii) LCM is the product of the highest powers of all factors, so LCM = 2 × 2 × 3 × 5 × 17 × 23 = 23460.
  why?
  LCM = the product of every prime factor that appears in any of the numbers, each taken with its greatest power.
8. (ii) Check: LCM × HCF = 23460 × 2 = 46920 and product of numbers = 510 × 92 = 46920.
  why?
  For any two positive whole numbers, HCF × LCM = the product of the two numbers, so this multiplication checks the HCF and LCM found.
9. (iii) Prime factorization of 336 is 2 × 2 × 2 × 2 × 3 × 7 and of 54 is 2 × 3 × 3 × 3.
  why?
  Every whole number greater than 1 is a product of primes in exactly one way (Fundamental Theorem of Arithmetic). Dividing again and again by a prime that goes in exactly finds that product.
10. (iii) HCF is the product of common factors, so HCF = 2 × 3 = 6.
  why?
  HCF = the product of the prime factors common to all the numbers, each taken with its smallest power.
11. (iii) LCM is the product of the highest powers of all factors, so LCM = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 7 = 3024.
  why?
  LCM = the product of every prime factor that appears in any of the numbers, each taken with its greatest power.
12. (iii) Check: LCM × HCF = 3024 × 6 = 18144 and product of numbers = 336 × 54 = 18144.
  why?
  For any two positive whole numbers, HCF × LCM = the product of the two numbers, so this multiplication checks the HCF and LCM found.
Answer: (i) HCF = 13, LCM = 182; (ii) HCF = 2, LCM = 23460; (iii) HCF = 6, LCM = 3024
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Ex 1.1 Q3
Use prime factorisation to find the LCM and HCF of: (i) 12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25.
1. (i) Prime factorisation: 12 = 2² × 3, 15 = 3 × 5, 21 = 3 × 7
  why?
  Every whole number greater than 1 is a product of primes in exactly one way (Fundamental Theorem of Arithmetic). Dividing again and again by a prime that goes in exactly finds that product.
2. (i) HCF is the product of common prime factors with lowest powers: 3
  why?
  HCF = the product of the prime factors common to all the numbers, each taken with its smallest power.
3. (i) LCM is the product of highest powers of all prime factors: 2² × 3 × 5 × 7 = 420
  why?
  LCM = the product of every prime factor that appears in any of the numbers, each taken with its greatest power.
4. (ii) Prime factorisation: 17, 23, and 29 are all prime numbers
  why?
  Every whole number greater than 1 is a product of primes in exactly one way (Fundamental Theorem of Arithmetic). Dividing again and again by a prime that goes in exactly finds that product.
5. (ii) HCF is 1 since there are no common prime factors
  why?
  HCF = the product of the prime factors common to all the numbers, each taken with its smallest power.
6. (ii) LCM is the product of all numbers: 17 × 23 × 29 = 11339
  why?
  LCM = the product of every prime factor that appears in any of the numbers, each taken with its greatest power.
7. (iii) Prime factorisation: 8 = 2³, 9 = 3², 25 = 5²
  why?
  Every whole number greater than 1 is a product of primes in exactly one way (Fundamental Theorem of Arithmetic). Dividing again and again by a prime that goes in exactly finds that product.
8. (iii) HCF is 1 since there are no common prime factors
  why?
  HCF = the product of the prime factors common to all the numbers, each taken with its smallest power.
9. (iii) LCM is the product of highest powers of all prime factors: 2³ × 3² × 5² = 8 × 9 × 25 = 1800
  why?
  LCM = the product of every prime factor that appears in any of the numbers, each taken with its greatest power.
Answer: (i) HCF = 3, LCM = 420; (ii) HCF = 1, LCM = 11339; (iii) HCF = 1, LCM = 1800
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Ex 1.1 Q4
HCF(306, 657) = 9. Find LCM(306, 657).
1. We use the property that the product of HCF and LCM of two numbers equals the product of the numbers.
  why?
  HCF = the product of the prime factors common to all the numbers, each taken with its smallest power.
2. So, HCF(306, 657) × LCM(306, 657) = 306 × 657.
  why?
  HCF = the product of the prime factors common to all the numbers, each taken with its smallest power.
3. Substitute the given HCF value: 9 × LCM(306, 657) = 306 × 657.
  why?
  HCF = the product of the prime factors common to all the numbers, each taken with its smallest power.
4. Calculate the product of the two numbers: 306 × 657 = 201042.
5. Now, LCM(306, 657) = 201042 ÷ 9.
  why?
  LCM = the product of every prime factor that appears in any of the numbers, each taken with its greatest power.
6. Perform the division: 201042 ÷ 9 = 22338.
Answer: LCM(306, 657) = 22338
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Ex 1.1 Q5
Can 6ⁿ end with the digit 0 for some natural number n?
1. For a number to end with the digit 0, it must be divisible by 10.
2. Since 10 = 2 × 5, the number must have both 2 and 5 as prime factors.
  why?
  Every whole number greater than 1 is a product of primes in exactly one way (Fundamental Theorem of Arithmetic). Dividing again and again by a prime that goes in exactly finds that product.
3. The prime factorization of 6 is 2 × 3.
  why?
  Every whole number greater than 1 is a product of primes in exactly one way (Fundamental Theorem of Arithmetic). Dividing again and again by a prime that goes in exactly finds that product.
4. Therefore, the prime factorization of 6ⁿ is (2 × 3)ⁿ = 2ⁿ × 3ⁿ.
  why?
  Every whole number greater than 1 is a product of primes in exactly one way (Fundamental Theorem of Arithmetic). Dividing again and again by a prime that goes in exactly finds that product.
5. The prime factors of 6ⁿ are only 2 and 3.
  why?
  Every whole number greater than 1 is a product of primes in exactly one way (Fundamental Theorem of Arithmetic). Dividing again and again by a prime that goes in exactly finds that product.
6. The prime factor 5 is missing in the factorization of 6ⁿ.
7. By the Fundamental Theorem of Arithmetic, the prime factorization of a number is unique.
  why?
  Every whole number greater than 1 is a product of primes in exactly one way (Fundamental Theorem of Arithmetic). Dividing again and again by a prime that goes in exactly finds that product.
8. Since 5 is not a factor of 6ⁿ, it cannot be divisible by 10.
9. Hence, 6ⁿ cannot end with the digit 0 for any natural number n.
Answer: No, 6ⁿ cannot end with the digit 0 for any natural number n.
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Ex 1.1 Q6
Why are 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 composite numbers?
1. (i) Take 13 common: 7 × 11 × 13 + 13 = 13 × (7 × 11 + 1) = 13 × 78 = 1014.
2. (i) 1014 has the factor 13, which is neither 1 nor 1014, so it is composite.
3. (ii) Take 5 common: 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 = 5 × (7 × 6 × 4 × 3 × 2 × 1 + 1) = 5 × 1009 = 5045.
4. (ii) 5045 has the factor 5, which is neither 1 nor 5045, so it is composite.
Answer: (i) 7 × 11 × 13 + 13 is composite; (ii) 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 is composite
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Ex 1.1 Q7
Sonia takes 18 minutes and Ravi 12 minutes for one round of a circular path. Starting together from the same point in the same direction, after how many minutes do they meet again at the start?
1. Sonia takes 18 minutes for one round and Ravi takes 12 minutes for one round.
2. They will meet again at the starting point after a time that is a multiple of both 18 and 12.
3. We need to find the Least Common Multiple (LCM) of 18 and 12.
  why?
  LCM = the product of every prime factor that appears in any of the numbers, each taken with its greatest power.
4. Prime factorization of 18 is 2 × 3².
  why?
  Every whole number greater than 1 is a product of primes in exactly one way (Fundamental Theorem of Arithmetic). Dividing again and again by a prime that goes in exactly finds that product.
5. Prime factorization of 12 is 2² × 3.
  why?
  Every whole number greater than 1 is a product of primes in exactly one way (Fundamental Theorem of Arithmetic). Dividing again and again by a prime that goes in exactly finds that product.
6. LCM is the product of the highest powers of all prime factors: 2² × 3².
  why?
  LCM = the product of every prime factor that appears in any of the numbers, each taken with its greatest power.
7. LCM = 4 × 9 = 36.
  why?
  LCM = the product of every prime factor that appears in any of the numbers, each taken with its greatest power.
8. Therefore, they meet again at the start after 36 minutes.
Answer: 36 minutes
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Other exercises in this chapter

Exercise 1.2 — 3 questions

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NCERT Class 10 Mathematics Exercise 1.1 Solutions: Real Numbers

Ex 1.1 Q1

Ex 1.1 Q2

Ex 1.1 Q3

Ex 1.1 Q4

Ex 1.1 Q5

Ex 1.1 Q6

Ex 1.1 Q7