Number Systems for CAT 2025: Concepts, Shortcuts, and Solved Examples

Basic Concepts

Explanation

Types of Numbers

• Natural Numbers (N): 1, 2, 3
• Whole Numbers (W): 0, 1, 2, 3
• Integers (Z): … –3, –2, –1, 0, 1, 2
• Rational Numbers (Q): Numbers of the form p/q (q ≠ 0), e.g. 2/3, –7/5.
• Irrational Numbers: Cannot be expressed as p/q, e.g. √2, π, e.
• Real Numbers (R): All rational + irrational numbers.
• Prime Numbers: Divisible only by 1 and itself. (e.g. 2, 3, 5, 7, 11…)
• Composite Numbers: Numbers with more than two factors.
• Even & Odd Numbers: Even = divisible by 2, Odd = not divisible by 2.

Divisibility Rules

• By 2: Last digit even.
• By 3: Sum of digits divisible by 3.
• By 4: Last two digits divisible by 4.
• By 5: Last digit 0 or 5.
• By 6: Divisible by 2 and 3.
• By 8: Last three digits divisible by 8.
• By 9: Sum of digits divisible by 9.
• By 11: Difference between sum of alternate digits divisible by 11.

HCF and LCM

• HCF: Product of common prime factors with the smallest power.
• LCM: Product of all prime factors with the highest power.
• Relation =

HCF X LCM = Product of Numbers

Remainders & Modular Arithmetic

• If N divided by m leaves remainder

R —> N ≡ r | M |

• (a + b) mod m = [(a mod m) + (b mod m)] mod m
• (a × b) mod m = [(a mod m) × (b mod m)] mod m
• Useful in cyclicity of digits, unit’s place, remainder questions.

Cyclicity & Unit’s Digit

• Cyclicity of powers helps find last digit quickly.
• For unit digit → check cycle length, use exponent mod cycle.

Number System Formula Sheet - PDF Download

Details

Particulars

Cumulative Property of Addition

A + B = B + A

Associative Property of Addition

(A + B) + C = A + (B + C)

Cumulative Property of Multiplication

A x B = B x A

Associative Property of Multiplication

(A x B) X C = A X (B x C)

Distributive Property of Multiplication with respect to Addition

A x (B + C) = (A x B) + (A x C)

Particulars

Details

a ⁿ x a ⁿ

a ^m+n

a ^m / a ⁿ

a ^m-n if m > n

a ^m / a ⁿ

1 / a ^n-m if m < n

a ^m / a ⁿ

a ⁰ = 1         if m = n

(a ^m ) ⁿ

a ^mn = (a ⁿ ) ^m