IPMAT Number System Preparation 2028 — Divisibility, Remainders, Factors & HCF-LCM

Why Number System is a High-Reward IPMAT Topic

Number System contributes 3 to 5 questions in IPMAT Indore and a similar proportion in Rohtak. What makes it special: a small set of concepts — divisibility, remainders, factors, and HCF-LCM — covers almost every Number System question ever asked in IPMAT. Students who invest 2 to 3 weeks in Number System consistently pick up 12 to 20 guaranteed marks.

Tanisq Chouhan, AIR 1 in JIPMAT, treated Number System as a high-reward topic from the start. “Number System questions look intimidating but most have elegant solutions. Once you know the 5 to 6 core concepts, you can crack any IPMAT Number System question in under 2 minutes,” he says.

Number System Topic Breakdown

Divisibility Rules — Must Memorise

Remainders — The Trickiest Number System Topic

Basic remainder: When N is divided by D, remainder = N − D × [N/D]. For IPMAT, you need three key remainder concepts:

Remainder of a sum: (A + B) mod D = ((A mod D) + (B mod D)) mod D. Remainder of a product: (A × B) mod D = ((A mod D) × (B mod D)) mod D. These two rules let you break down complex remainder problems.

Fermat's Little Theorem: If p is prime and gcd(a, p) = 1, then a^(p−1) ≡ 1 (mod p). This simplifies remainder of large power problems. Example: Find remainder when 2^100 is divided by 101. Since 101 is prime and gcd(2,101) = 1, 2^100 ≡ 1 (mod 101). Answer: 1.

Factors & Number of Divisors

To find the number of factors: Express N as a product of prime powers: N = p¹^a × p²^b × p³^c. Number of factors = (a+1)(b+1)(c+1). Example: 360 = 2³ × 3² × 5¹. Factors = (3+1)(2+1)(1+1) = 24.

Sum of all factors: For N = p¹^a × p²^b, sum of factors = (p¹^0 + p¹^1 + ... + p¹^a) × (p²^0 + p²^1 + ... + p²^b). IPMAT loves asking: how many factors of N are perfect squares, perfect cubes, or odd numbers.

HCF & LCM

Key relationship: HCF × LCM = Product of two numbers (valid for exactly 2 numbers only). For HCF: use prime factorisation, take lowest powers of common primes. For LCM: take highest powers of all primes.

Application questions: Largest tile that can cover a floor of dimensions a × b = HCF(a, b). Smallest container that holds exact whole numbers of items of sizes a and b = LCM(a, b). Bells ringing at intervals of a and b minutes — next simultaneous ring = LCM(a, b) minutes.

Unit Digit & Cyclicity

The unit digit of powers follows a cycle. Cycles to memorise: 2 → cycle of 4 (2,4,8,6). 3 → cycle of 4 (3,9,7,1). 4 → cycle of 2 (4,6). 7 → cycle of 4 (7,9,3,1). 8 → cycle of 4 (8,4,2,6). 9 → cycle of 2 (9,1). 1, 5, 6 → always 1, 5, 6 regardless of power.

To find unit digit of a^n: find n mod (cycle length). If the cycle of a is 4, find n mod 4. The result gives the position in the cycle. Example: unit digit of 7^53 — cycle is (7,9,3,1), 53 mod 4 = 1, so unit digit = 7.

Frequently Asked Questions — IPMAT Number System

How many Number System questions appear in IPMAT?

3 to 5 questions in IPMAT Indore and similar in Rohtak. Trending slightly upward in recent years.

What is the divisibility rule for 11?

Alternating sum of digits (digits at odd positions minus digits at even positions from right) is divisible by 11.

How do I find the number of factors of a number?

Express as prime factorisation N = p1^a × p2^b × p3^c. Number of factors = (a+1)(b+1)(c+1).

What is Fermat's Little Theorem?

If p is prime and gcd(a, p) = 1, then a^(p-1) ≡ 1 (mod p). Used to find remainders of large power divisions.

What is the HCF × LCM relationship?

HCF × LCM = product of two numbers. Valid for exactly two numbers only.

How do I find the unit digit of a large power?

Find the cyclicity of the base digit, then find power mod cycle length to locate position in the cycle.

How long should I spend on Number System preparation?

2 to 3 weeks. Cover all 6 core topics, solve 20 questions per topic, then mix with previous year questions.

Is Number System asked in IPMAT SA section?

Yes. IIM Indore SA section includes Number System questions. With no negative marking on SA, attempt every Number System SA question.

What is cyclicity and why is it important?

Cyclicity is the repeating pattern of unit digits in powers. It lets you find the unit digit of any power in seconds without actual multiplication.

What is the remainder when a sum is divided?

(A+B) mod D = ((A mod D) + (B mod D)) mod D. Apply this rule to break down complex remainder expressions.

Can arts students score in Number System?

Yes. Number System does not require advanced maths. Divisibility, HCF-LCM, and factors are Class 6 to 8 level concepts applied with clever questions.

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