In-Depth: Factors and Multiples

CLAT Application & Relevance

Importance: Low. While not directly a core focus for CLAT QT, understanding factors and multiples is foundational. It underpins concepts like divisibility rules, LCM, and HCF, which can indirectly aid in simplifying calculations or solving specific types of word problems within Data Interpretation sets. Questions directly asking for factors/multiples are rare, but the concepts are implicitly applied.

How it's tested: Implicitly, in questions requiring prime factorization for LCM/HCF; finding numbers based on certain properties (e.g., numbers that are multiples of X and Y); simplifying ratios and fractions.

Section 1: Core Concepts & Prime Factorization

Factors are numbers that divide another number completely (without leaving a remainder). Multiples are numbers that you get when you multiply a number by any positive integer.

Key Definitions

Factor: If a number 'a' divides another number 'b' exactly, then 'a' is a factor of 'b'. Every number has at least two factors: 1 and itself.
Multiple: A multiple of a number is the product of that number and any positive integer. Multiples of a number are infinite.
Prime Factor: A factor that is also a prime number.

Prime Factorization Method

This is the process of breaking down a composite number into its prime factors. It's essential for understanding LCM, HCF, and the number of factors.

Steps:

Start dividing the number by the smallest prime number (2, 3, 5, 7...).
Continue dividing the quotient by prime numbers until the quotient is 1.
The product of these prime divisors is the prime factorization of the original number.

Example: Prime Factorization of 60

        60 ÷ 2 = 30
        30 ÷ 2 = 15
        15 ÷ 3 = 5
        5 ÷ 5 = 1
        So, 60 = 2 × 2 × 3 × 5 = 2² × 3¹ × 5¹

Properties related to Factors

Number of Factors: If a number N = p1^a * p2^b * p3^c * ... (where p1, p2, p3 are prime factors), then the total number of factors of N is (a+1)(b+1)(c+1)...
Sum of Factors: For N = p1^a * p2^b * ..., Sum = (p1^(a+1)-1)/(p1-1) * (p2^(b+1)-1)/(p2-1) * ...

Section 2: Solved CLAT-Style Examples (Application)

Example 1: Identifying Factors and Multiples from a Set

Passage Context: "A legal database stores case numbers. The valid case numbers must be factors of 96. Also, for a specific analysis, case numbers that are multiples of 8 are filtered."

Question A: "List all positive factors of 96."

Question B: "From the factors of 96, identify which ones are also multiples of 8."

Detailed Solution A (Factors of 96):
1. Find pairs of numbers that multiply to 96: 1 × 96 2 × 48 3 × 32 4 × 24 6 × 16 8 × 12
2. List all unique factors: {1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96}.
Answer A: The positive factors of 96 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96.

Detailed Solution B (Factors that are Multiples of 8):
1. Recall Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96...
2. Cross-reference with factors of 96: From the list of factors: 8, 12, 16, 24, 32, 48, 96. Check which of these are divisible by 8: 8, 16, 24, 32, 48, 96.
Answer B: The factors of 96 that are also multiples of 8 are 8, 16, 24, 32, 48, and 96.

Example 2: Counting the Number of Factors (Advanced)

Passage Context: "A special legal code is determined by the number of divisors (factors) a specific integer has. For a particular case, this integer is 720."

Question: "How many total factors does the number 720 have?"

Detailed Solution:
1. Perform Prime Factorization of 720: 720 = 72 × 10 72 = 8 × 9 = 2³ × 3² 10 = 2 × 5 So, 720 = 2³ × 3² × 2¹ × 5¹ = 2^(3+1) × 3² × 5¹ = 2⁴ × 3² × 5¹
2. Identify Powers of Prime Factors: p1 = 2, a = 4 p2 = 3, b = 2 p3 = 5, c = 1
3. Apply Formula for Number of Factors: (a+1)(b+1)(c+1) Number of factors = (4+1)(2+1)(1+1) = (5)(3)(2) = 30.
Answer: The number 720 has 30 factors.