Real Numbers | Exercise 1.1

Question 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

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Solution
Understand the Question
  • To find the HCF and LCM of integers using the prime factorisation method:
    • Express each number as a product of prime powers.
    • HCF: Product of the smallest power of each common prime factor involved in the numbers.
    • LCM: Product of the greatest power of each prime factor involved in the numbers.
  • If there are no common prime factors among the numbers, the HCF\text{HCF} is 11.

(i) 12, 15 and 21

Step 1 · Find Prime Factorisation, HCF, and LCM

Prime factorisation of the given numbers:Diagram 1

12=2×2×3=22×3115=3×5=31×5121=3×7=31×71\begin{aligned} 12 &= 2 \times 2 \times 3 = 2^2 \times 3^1 \\ 15 &= 3 \times 5 = 3^1 \times 5^1 \\ 21 &= 3 \times 7 = 3^1 \times 7^1 \end{aligned}

The lowest power of the common prime factor 33 gives: HCF(12,15,21)=3\text{HCF}(12, 15, 21) = 3

Taking the highest power of all prime factors involved (22,31,51,712^2, 3^1, 5^1, 7^1):

LCM(12,15,21)=22×31×51×71=4×3×5×7=12×35=420\begin{aligned} \text{LCM}(12, 15, 21) &= 2^2 \times 3^1 \times 5^1 \times 7^1 \\ &= 4 \times 3 \times 5 \times 7 \\ &= 12 \times 35 \\ &= 420 \end{aligned}
Answer

(i) HCF=3,LCM=420\text{HCF} = 3, \quad \text{LCM} = 420

(ii) 17, 23 and 29

Step 1 · Find Prime Factorisation, HCF, and LCM

Since 1717, 2323, and 2929 are all prime numbers:

17=17123=23129=291\begin{aligned} 17 &= 17^1 \\ 23 &= 23^1 \\ 29 &= 29^1 \end{aligned}

There are no common prime factors other than 11: HCF(17,23,29)=1\text{HCF}(17, 23, 29) = 1

Taking the product of all prime factors:

LCM(17,23,29)=17×23×29=391×29=11339\begin{aligned} \text{LCM}(17, 23, 29) &= 17 \times 23 \times 29 \\ &= 391 \times 29 \\ &= 11339 \end{aligned}
Answer

(ii) HCF=1,LCM=11339\text{HCF} = 1, \quad \text{LCM} = 11339

(iii) 8, 9 and 25

Step 1 · Find Prime Factorisation, HCF, and LCM

Prime factorisation of the given numbers:

8=2×2×2=239=3×3=3225=5×5=52\begin{aligned} 8 &= 2 \times 2 \times 2 = 2^3 \\ 9 &= 3 \times 3 = 3^2 \\ 25 &= 5 \times 5 = 5^2 \end{aligned}

There are no common prime factors among the numbers: HCF(8,9,25)=1\text{HCF}(8, 9, 25) = 1

Taking the highest power of all prime factors involved (23,32,522^3, 3^2, 5^2):

LCM(8,9,25)=23×32×52=8×9×25=72×25=1800\begin{aligned} \text{LCM}(8, 9, 25) &= 2^3 \times 3^2 \times 5^2 \\ &= 8 \times 9 \times 25 \\ &= 72 \times 25 \\ &= 1800 \end{aligned}
Answer

(iii) HCF=1,LCM=1800\text{HCF} = 1, \quad \text{LCM} = 1800

Common Mistakes
  • HCF of Co-prime Numbers: When numbers have no common prime factors, their HCF\text{HCF} is 11, not 00.
  • Three-Number Property Fallacy: The relation HCF(a,b,c)×LCM(a,b,c)=a×b×c\text{HCF}(a, b, c) \times \text{LCM}(a, b, c) = a \times b \times c is false for three numbers; it only holds true for two numbers.
  • Power Confusion: Confusing lowest powers (used for common factors in HCF) with highest powers of all factors (used for LCM).

More questions in Exercise 1.1

Q1

Express each number as a product of its prime factors:

(i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429

Q2

Find the LCM and HCF of the following pairs of integers and verify that LCM×HCF=product of the two numbers\text{LCM} \times \text{HCF} = \text{product of the two numbers}.

(i) 26 and 91

(ii) 510 and 92

(iii) 336 and 54

Q3

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

Q4

Given that HCF(306,657)=9\text{HCF}(306, 657) = 9, find LCM(306,657)\text{LCM}(306, 657).

Q5

Check whether 6n6^n can end with the digit 0 for any natural number nn.

Q6

Explain why 7×11×13+137 \times 11 \times 13 + 13 and 7×6×5×4×3×2×1+57 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5 are composite numbers.

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?

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