Match List I with List IIList IList II(A) LCM of 22 × 33 × 54 × 7 and 23 × 3 × 5 × 74(I) 26 × 74 (B) HCF of 23 × 32 × 5 and 32 × 24 × 53(II) 212 × 35 × 57 × 714(C) LCM of 57 × 75 × 212 and 35 × 714 × 26(III) 23 × 33 × 54 × 74(D) HCF of 57 × 75 × 212 and 35 × 74 × 26 × 11(IV) 23 × 32 × 5Choose the correct answer from the options given below:

Question

Prepp Team · Accepted Answer

Calculating LCM and HCF using Prime Factorization

To solve this problem, we need to understand how to calculate the Least Common Multiple (LCM) and Highest Common Factor (HCF) of numbers when they are given in their prime factorized form. Let's analyze each part of List I and calculate the required LCM or HCF.

Understanding LCM and HCF

When numbers are expressed as a product of their prime factors:

The LCM is found by taking the highest power of all the prime factors that appear in any of the given numbers and multiplying them together.
The HCF (also known as GCD - Greatest Common Divisor) is found by taking the lowest power of only the common prime factors that appear in all the given numbers and multiplying them together. If a prime factor is not common to all numbers, it is not included in the HCF.

Step-by-Step Calculation for Each Item in List I

(A) LCM of \(2^2 \times 3^3 \times 5^4 \times 7\) and \(2^3 \times 3 \times 5 \times 7^4\)

The two numbers are \(N_1 = 2^2 \times 3^3 \times 5^4 \times 7^1\) and \(N_2 = 2^3 \times 3^1 \times 5^1 \times 7^4\). Let's look at the powers of each prime factor present:

Prime factor 2: The powers are \(2^2\) and \(2^3\). The highest power is \(2^3\).
Prime factor 3: The powers are \(3^3\) and \(3^1\). The highest power is \(3^3\).
Prime factor 5: The powers are \(5^4\) and \(5^1\). The highest power is \(5^4\).
Prime factor 7: The powers are \(7^1\) and \(7^4\). The highest power is \(7^4\).

Therefore, the LCM is \(2^3 \times 3^3 \times 5^4 \times 7^4\). This matches option (III) in List II.

(B) HCF of \(2^3 \times 3^2 \times 5\) and \(3^2 \times 2^4 \times 5^3\)

The two numbers are \(N_3 = 2^3 \times 3^2 \times 5^1\) and \(N_4 = 2^4 \times 3^2 \times 5^3\). Let's look at the powers of the common prime factors:

Prime factor 2: Common in both. The powers are \(2^3\) and \(2^4\). The lowest power is \(2^3\).
Prime factor 3: Common in both. The powers are \(3^2\) and \(3^2\). The lowest power is \(3^2\).
Prime factor 5: Common in both. The powers are \(5^1\) and \(5^3\). The lowest power is \(5^1\).

Therefore, the HCF is \(2^3 \times 3^2 \times 5\). This matches option (IV) in List II.

(C) LCM of \(5^7 \times 7^5 \times 2^{12}\) and \(3^5 \times 7^{14} \times 2^6\)

The two numbers are \(N_5 = 2^{12} \times 5^7 \times 7^5\) and \(N_6 = 2^6 \times 3^5 \times 7^{14}\). Let's look at the powers of each prime factor present in either number:

Prime factor 2: The powers are \(2^{12}\) and \(2^6\). The highest power is \(2^{12}\).
Prime factor 3: Present as \(3^5\) in \(N_6\). Highest power is \(3^5\).
Prime factor 5: Present as \(5^7\) in \(N_5\). Highest power is \(5^7\).
Prime factor 7: The powers are \(7^5\) and \(7^{14}\). The highest power is \(7^{14}\).

Therefore, the LCM is \(2^{12} \times 3^5 \times 5^7 \times 7^{14}\). This matches option (II) in List II.

(D) HCF of \(5^7 \times 7^5 \times 2^{12}\) and \(3^5 \times 7^4 \times 2^6 \times 11\)

The two numbers are \(N_7 = 2^{12} \times 5^7 \times 7^5\) and \(N_8 = 2^6 \times 3^5 \times 7^4 \times 11^1\). Let's look at the powers of the common prime factors:

Prime factor 2: Common in both. The powers are \(2^{12}\) and \(2^6\). The lowest power is \(2^6\).
Prime factor 3: Not common.
Prime factor 5: Not common.
Prime factor 7: Common in both. The powers are \(7^5\) and \(7^4\). The lowest power is \(7^4\).
Prime factor 11: Not common.

Therefore, the HCF is \(2^6 \times 7^4\). This matches option (I) in List II.

Summary of Matches

List I	Calculation	Result	Matches List II
(A)	LCM of \(2^2 \times 3^3 \times 5^4 \times 7\) and \(2^3 \times 3 \times 5 \times 7^4\)	\(2^3 \times 3^3 \times 5^4 \times 7^4\)	(III)
(B)	HCF of \(2^3 \times 3^2 \times 5\) and \(3^2 \times 2^4 \times 5^3\)	\(2^3 \times 3^2 \times 5\)	(IV)
(C)	LCM of \(5^7 \times 7^5 \times 2^{12}\) and \(3^5 \times 7^{14} \times 2^6\)	\(2^{12} \times 3^5 \times 5^7 \times 7^{14}\)	(II)
(D)	HCF of \(5^7 \times 7^5 \times 2^{12}\) and \(3^5 \times 7^4 \times 2^6 \times 11\)	\(2^6 \times 7^4\)	(I)

So the correct matching is (A)-(III), (B)-(IV), (C)-(II), (D)-(I).

Checking the Options

Let's compare our results to the given options:

Option (a): (A)-(III), (B)-(II), (C)-(IV), (D)-(I) - Incorrect (B and C don't match)
Option (b): (A)-(IV), (B)-(III), (C)-(I), (D)-(II) - Incorrect (A, B, C, D don't match)
Option (c): (A)-(III), (B)-(IV), (C)-(II), (D)-(I) - Correct
Option (d): (A)-(III), (B)-(IV), (C)-(I), (D)-(II) - Incorrect (C and D are swapped)

Based on our calculations, option (c) provides the correct set of matches.

Revision Table: LCM and HCF Properties

Concept	Definition	How to calculate from prime factors	Example (Numbers: \(2^2 \times 3^1\), \(2^1 \times 3^3\))
LCM (Least Common Multiple)	The smallest positive integer that is a multiple of two or more numbers.	Take the highest power of all prime factors present in any number.	LCM(\(2^2 \times 3^1\), \(2^1 \times 3^3\)) = \(2^{\text{max}(2,1)} \times 3^{\text{max}(1,3)} = 2^2 \times 3^3\)
HCF (Highest Common Factor) / GCD (Greatest Common Divisor)	The largest positive integer that divides two or more numbers without leaving a remainder.	Take the lowest power of only the common prime factors present in all numbers.	HCF(\(2^2 \times 3^1\), \(2^1 \times 3^3\)) = \(2^{\text{min}(2,1)} \times 3^{\text{min}(1,3)} = 2^1 \times 3^1\)

Additional Information: Applications of LCM and HCF

Understanding LCM and HCF is fundamental in number theory and has various applications:

Fractions: LCM is used to find the least common denominator (LCD) when adding or subtracting fractions.
Time and Cycles: LCM can be used to find when events that repeat at different intervals will occur simultaneously (e.g., buses arriving at a stop, clocks chiming).
Dividing Quantities: HCF is used to divide quantities into the largest possible equal parts (e.g., distributing items into groups).
Simplifying Fractions: HCF is used to simplify fractions to their lowest terms by dividing both the numerator and the denominator by their HCF.
Relationship between LCM and HCF: For any two positive integers \(a\) and \(b\), the product of their LCM and HCF is equal to the product of the numbers themselves, i.e., LCM(\(a, b\)) \(\times\) HCF(\(a, b\)) = \(a \times b\). This relationship is useful for finding one value if the other two are known.

Prepp Team · Answer

(a) (A)-(III), (B)-(II), (C)-(IV), (D)-(I)

Prepp Team · Answer

(b) (A)-(IV), (B)-(III), (C)-(I), (D)-(II)

Prepp Team · Answer

(d) (A)-(III), (B)-(IV), (C)-(I), (D)-(II)

List I	List II
(A) LCM of 2² × 3³ × 5⁴ × 7 and 2³ × 3 × 5 × 7⁴	(I) 2⁶ × 7⁴
(B) HCF of 2³ × 3² × 5 and 3² × 2⁴ × 5³	(II) 2¹² × 3⁵ × 5⁷ × 7¹⁴
(C) LCM of 5⁷ × 7⁵ × 2¹² and 3⁵ × 7¹⁴ × 2⁶	(III) 2³ × 3³ × 5⁴ × 7⁴
(D) HCF of 5⁷ × 7⁵ × 2¹² and 3⁵ × 7⁴ × 2⁶ × 11	(IV) 2³ × 3² × 5