Exercise 1.1 answers: Real Numbers

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(i) 140

1. 140 is even, so divide by 2: 140 ÷ 2 = 70.
2. 70 is even too: 70 ÷ 2 = 35.
3. 35 is not even, but it ends in 5, so divide by 5: 35 ÷ 5 = 7.
4. 7 is prime, so stop here.

Answer140 = 2² × 5 × 7

(ii) 156

1. 156 is even: 156 ÷ 2 = 78.
2. 78 is even too: 78 ÷ 2 = 39.
3. 39 is not even. Its digits add to 12, which 3 divides: 39 ÷ 3 = 13.
4. 13 is prime, so stop here.

Answer156 = 2² × 3 × 13

(iii) 3825

1. 3825 is odd. Its digits add to 18, which 3 divides: 3825 ÷ 3 = 1275.
2. 1275's digits add to 15, again divisible by 3: 1275 ÷ 3 = 425.
3. 425 ends in 5: 425 ÷ 5 = 85.
4. 85 ÷ 5 = 17, and 17 is prime, so stop here.

Answer3825 = 3² × 5² × 17

(iv) 5005

1. 5005 ends in 5: 5005 ÷ 5 = 1001.
2. 1001 ÷ 7 = 143.
3. 143 ÷ 11 = 13.
4. 13 is prime, so stop here.

Answer5005 = 5 × 7 × 11 × 13

(v) 7429

1. 7429 is odd, its digits do not add to a multiple of 3, and it does not end in 0 or 5, so 2, 3 and 5 are ruled out.
2. Divide by 17: 7429 ÷ 17 = 437.
3. 437 ÷ 19 = 23.
4. 23 is prime, so stop here.

Answer7429 = 17 × 19 × 23

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(i) 26 and 91

1. Write both as products of primes: 26 = 2 × 13, 91 = 7 × 13.
2. HCF takes the lower power of every common prime: only 13 is common, so HCF = 13.
3. LCM takes the higher power of every prime present: LCM = 2 × 7 × 13 = 182.
4. Check: LCM × HCF = 182 × 13 = 2366, and 26 × 91 = 2366 too. They match.

AnswerHCF = 13, LCM = 182

(ii) 510 and 92

1. 510 = 2 × 3 × 5 × 17, 92 = 2² × 23.
2. Only 2 is common to both, at the lower power 2¹, so HCF = 2.
3. LCM = 2² × 3 × 5 × 17 × 23 = 23460.
4. Check: LCM × HCF = 23460 × 2 = 46920, and 510 × 92 = 46920 too. They match.

AnswerHCF = 2, LCM = 23460

(iii) 336 and 54

1. 336 = 2⁴ × 3 × 7, 54 = 2 × 3³.
2. Common primes are 2 and 3; take the lower power of each: 2¹ × 3¹ = 6, so HCF = 6.
3. LCM takes the higher power of each prime present: 2⁴ × 3³ × 7 = 3024.
4. Check: LCM × HCF = 3024 × 6 = 18144, and 336 × 54 = 18144 too. They match.

AnswerHCF = 6, LCM = 3024

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(i) 12, 15 and 21

1. 12 = 2² × 3, 15 = 3 × 5, 21 = 3 × 7.
2. Only 3 is common to all three, so HCF = 3.
3. LCM takes the highest power of every prime that appears: 2² × 3 × 5 × 7 = 420.

AnswerHCF = 3, LCM = 420

(ii) 17, 23 and 29

1. 17, 23 and 29 are all prime, and all different from each other.
2. No prime is common to all three, so HCF = 1.
3. Since nothing is shared, LCM is just their product: 17 × 23 × 29 = 11339.

AnswerHCF = 1, LCM = 11339

(iii) 8, 9 and 25

1. 8 = 2³, 9 = 3², 25 = 5² — each number is built from a different prime.
2. No prime is common to all three, so HCF = 1.
3. LCM is the product of the highest powers seen: 2³ × 3² × 5² = 1800.

AnswerHCF = 1, LCM = 1800

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1. For any two numbers, HCF × LCM = product of the two numbers.
2. 306 × 657 = 201042.
3. LCM = 201042 ÷ 9 = 22338.

Answer22338

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1. A number ends in the digit 0 exactly when it is divisible by 10, and 10 = 2 × 5.
2. So 6ⁿ could end in 0 only if 5 divides 6ⁿ.
3. 6 = 2 × 3, so 6ⁿ = 2ⁿ × 3ⁿ — its only prime factors are 2 and 3.
4. By the uniqueness of prime factorisation, 5 is never a factor of 6ⁿ, for any n.
5. So 6ⁿ can never end in the digit 0.

AnswerNo, 6ⁿ can never end with the digit 0.

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Open NCERT p. 5One way to think about it

1. 7 × 11 × 13 + 13 = 13 × (7 × 11 + 1) = 13 × 78, since 13 is a common factor of both terms.
2. 13 × 78 is a product of two whole numbers, each bigger than 1, so it is composite.
3. 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040, and every factor from 1 to 7 is included, so 5 is a common factor of 5040 and of the 5 being added: 5040 + 5 = 5 × (1008 + 1) = 5 × 1009.
4. 5 × 1009 is also a product of two whole numbers, each bigger than 1, so it too is composite.

In shortBoth numbers have a common factor that can be taken out (13 in the first, 5 in the second), leaving a product of two numbers greater than 1 — so both are composite.

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1. Sonia is back at the starting point after every multiple of 18 minutes; Ravi is back after every multiple of 12 minutes.
2. They are both at the start together for the first time after the smallest common multiple of 18 and 12 — that is their LCM.
3. 18 = 2 × 3², 12 = 2² × 3, so LCM = 2² × 3² = 36.

Answer36 minutes

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