📘 Properties of Numbers (CAT Quant)

The Properties of Numbers form the foundation of the Number System and are frequently tested in CAT through logic-based, remainder-based, and divisibility-based questions. CAT rarely asks direct formula questions; instead, it checks conceptual clarity and pattern recognition.

1️⃣ Classification of Numbers

Natural Numbers

Positive counting numbers: 1, 2, 3, …

Whole Numbers

0 and all natural numbers.

Integers

…, −3, −2, −1, 0, 1, 2, 3, …

Rational Numbers

Numbers that can be expressed as p/q, where q ≠ 0.

Irrational Numbers

Numbers that cannot be expressed as a fraction (e.g., √2, π).

🔹 CAT Focus: Identifying the type of number involved to apply the correct logic.

2️⃣ Even and Odd Numbers

  • Even number: Divisible by 2

  • Odd number: Not divisible by 2

Key Properties:

  • Even ± Even = Even

  • Odd ± Odd = Even

  • Even ± Odd = Odd

  • Even × Any = Even

  • Odd × Odd = Odd

🔹 CAT Application:
Used in parity-based elimination, especially in TITA questions.

3️⃣ Prime and Composite Numbers

Prime Number

A number greater than 1 with exactly two factors: 1 and itself.

Composite Number

A number with more than two factors.

Important Points:

  • 2 is the only even prime number

  • 1 is neither prime nor composite

🔹 CAT Focus:
Questions involving factor count, divisibility, and LCM–HCF logic.

4️⃣ Divisibility Rules (Conceptual Use)

Some commonly used divisibility checks:

  • Divisible by 2 → Last digit even

  • Divisible by 3 → Sum of digits divisible by 3

  • Divisible by 9 → Sum of digits divisible by 9

  • Divisible by 11 → Difference of alternating digit sums is 0 or a multiple of 11

🔹 CAT Tip:
Used for fast elimination, not lengthy calculations.

5️⃣ Factors and Multiples

Factor

A number that divides another number exactly.

Multiple

A number obtained by multiplying a given number by an integer.

Key Observations:

  • Every number has at least two factors: 1 and itself

  • Prime numbers have exactly two factors

  • Number of factors depends on prime factorization

🔹 CAT Focus:
Factor-based questions often combine remainder, divisibility, and LCM/HCF concepts.

6️⃣ Properties of Zero and One

Zero (0)

  • 0 multiplied by any number = 0

  • Division by 0 is undefined

One (1)

  • Multiplicative identity

  • Does not change the value when multiplied

🔹 CAT Trap:
Division involving zero and assumptions about factors of 1.

7️⃣ Positive and Negative Numbers

  • Product of two positive numbers → Positive

  • Product of two negative numbers → Positive

  • Product of positive and negative → Negative

🔹 CAT Use:
Sign-based logic in equations, inequalities, and remainder problems.

8️⃣ Absolute Value

Absolute value of a number is its distance from zero, always non-negative.

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