Number System

Number System

Concept

The Number System is the backbone of all quantitative reasoning in CUET. At its heart are two ideas: HCF (Highest Common Factor) and LCM (Lowest Common Multiple). Think of HCF as the biggest number that divides both numbers evenly — like the largest tile that would fit perfectly on a floor of dimensions a × b. The LCM is the smallest number that both numbers divide into — like the earliest time two traffic lights with different cycle times will flash green together.

Divisibility rules are shortcuts that let you check if a number is divisible by 2, 3, 4, 5, and so on without actually dividing. These save precious seconds in the exam. For instance, 1,38,600 is divisible by 8? Check the last three digits — 600 ÷ 8 = 75 with no remainder. Done. No long division needed.

Remainders appear when division isn’t clean. If 73 ÷ 4 gives remainder 1, it means 73 = 4 × 18 + 1. Remainders have fascinating properties — adding or multiplying numbers and then finding a remainder often gives the same answer as finding remainders first.

Key Formulas

Formula	Use
HCF × LCM = a × b	Relates HCF and LCM of two numbers a and b
HCF = product of common prime factors (lowest powers)	Prime factorisation method
LCM = product of highest powers of all prime factors	Prime factorisation method
Dividend = Divisor × Quotient + Remainder	Basic remainder formula
(a + b) mod n = [(a mod n) + (b mod n)] mod n	Remainder addition property

Worked Example

Q: Find the HCF and LCM of 18 and 30 using prime factorisation.

Step 1: Prime factorise both numbers.

• 18 = 2 × 3²
• 30 = 2 × 3 × 5

Step 2: HCF = product of common primes with lowest powers = 2¹ × 3¹ = 6

Step 3: LCM = product of highest powers of all primes = 2¹ × 3² × 5¹ = 90

Step 4: Verify: HCF × LCM = 6 × 90 = 540 = 18 × 30 ✓

Answer: HCF = 6, LCM = 90

Common Errors

• Confusing HCF with LCM (HCF is the smaller number, LCM is larger) → Draw a factor Venn diagram to visualise
• Forgetting to take the lowest power of common primes when finding HCF → always write primes in exponent form first
• Applying remainder addition incorrectly — (a × b) mod n ≠ (a mod n) × (b mod n) in general → use: (a × b) mod n = [(a mod n) × (b mod n)] mod n