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REAL NUMBERS – EXERCISE 1.1 (Detailed Solutions)

3. 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
Ans :- LCM & HCF using prime factorisation
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(i) 12, 15 and 21
12 = 2² × 3
15 = 3 × 5
21 = 3 × 7
HCF = 1
LCM = 2² × 3 × 5 × 7 = 420
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(ii) 17, 23 and 29
All are primes.
HCF = 1
LCM = 17 × 23 × 29 = 11339
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(iii) 8, 9 and 25
8 = 2³
9 = 3²
25 = 5²
HCF = 1
LCM = 2³ × 3² × 5² = 1800
Jaha sabhi numbers ka HCF = 1 aata hai, waha numbers ek dusre ke co-prime hote hain.

4. Given that HCF (306, 657) = 9, find LCM (306, 657).
Ans :- Given HCF (306, 657) = 9, find LCM (306, 657)
We know:
LCM × HCF = 306 × 657
So,
LCM = (306 × 657) ÷ 9
LCM = (201,042 ÷ 9)
LCM = 22338

5. Check whether 6n can end with the digit 0 for any natural number n.
Ans :- 6ⁿ always ends with 6 (because 6 × 6 = 36 → ends with 6).
To end with 0, a number must have factors 2 and 5.
6ⁿ = (2 × 3)ⁿ has no factor 5, so:
6ⁿ can never end with digit 0.

6. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.
Ans :-
(i) 7 × 11 × 13 + 13
= 13(7 × 11 + 1)
= 13(77 + 1)
= 13 × 78
Product of two integers → composite.
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(ii) 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5
7! + 5
= 5040 + 5
= 5045
= 5 × 1009
Since it has factors → composite.

7. 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?
Ans :- A circular track: Sonia = 18 min, Ravi = 12 min
They meet at starting point after LCM(18, 12):
Prime factors:
18 = 2 × 3²
12 = 2² × 3
LCM = 2² × 3² = 36
They meet after 36 minutes.