🔢 Quant

Number System

Remainders, divisibility, factors, LCM/HCF, base conversions. Often 4–6 questions in CAT.

of Quant

Why This Topic Matters

Total PYQs📊

of 1002 · 2021–2025

Years featured📅

5/5

of recent CAT years

% of Quant📈

~9%

of section questions

Est. hours⏱️

~15h

to master

~3/22

2021

~2/22

2022

~3/22

2023

~2/22

2024

~3/22

2025

Small topic, big leverage

Number System is roughly 2 of the 22 Quant questions per slot — but it is conceptually dense, and a handful of results unlock a disproportionate number of questions. The page-level sub-topics are Divisibility & Remainders and Factors, LCM/HCF, Bases & Digits; below are the cross-cutting essentials, including the modular tools CAT loves.

What CAT 2021–2025 actually asked

Sub-skill	2021	2022	2023	2024	2025	Avg/slot
Number properties & digits	2.0	–	0.7	–	1.0	0.7
Remainders & divisibility	–	1.0	0.3	1.3	0.3	0.6
Factors, LCM & HCF	–	0.7	1.3	–	1.0	0.6
Indices & surds (NS-style)	0.3	–	–	–	–	0.1

🎯PYQ Evidence

Number System alternates its face but never disappears. Some flavour of NS appeared in every year of 2021–2025, but the flavour rotates: 2021 leaned on number properties and digits (~2 per slot), 2022 and 2024 on remainders/divisibility, 2023 and 2025 on factors and LCM–HCF. Practical reading: prepare all three pillars — properties/digits, remainders, factors — because skipping whichever one "didn't come last year" is exactly how it catches people.

Factors and multiples

If $N=p^{a}\,q^{b}\,r^{c}$ (prime factorisation), then:

Number of factors $=(a+1)(b+1)(c+1)$
Product of all factors $=N^{\,t/2}$ , where $t$ is the number of factors
HCF × LCM $=$ product of the two numbers: $\ \text{HCF}(a,b)\times\text{LCM}(a,b)=a\times b$

Divisibility shortcuts:

Divisor	Test
3 / 9	digit sum divisible by 3 / 9
4	last two digits divisible by 4
8	last three digits divisible by 8
11	alternating digit sum divisible by 11

Remainders the modular way

Work with residues, not whole numbers. Sums and products of remainders give the remainder of the sum/product, so reduce the base before powering.

Power cycles: the units digit of $a^n$ repeats with a period that divides 4 — so take the exponent $\bmod\,4$ .
Fermat's Little Theorem: for a prime $p$ with $p\nmid a$ , $\ a^{\,p-1}\equiv 1 \pmod{p}$ .
Chinese Remainder idea: coprime moduli give a unique solution modulo their product — solve each congruence, then stitch.

Worked examples

✏️Worked Example

Units digit of $7^{2024}$ . The cycle of $7$ is $7,9,3,1$ (period 4). Since $2024\equiv 0\pmod 4$ , we land on the last term of the cycle: units digit $=\mathbf{1}$ .

How many factors does $360$ have? $360=2^{3}\cdot3^{2}\cdot5^{1}$ , so the count is $(3+1)(2+1)(1+1)=4\cdot3\cdot2=\mathbf{24}$ .

Remainder of $43\times 47$ on division by 5. Reduce first: $43\equiv 3$ , $47\equiv 2$ , so $43\cdot47\equiv 3\cdot2=6\equiv\mathbf{1}\pmod 5$ .

⚠️CAT Trap

"Exponent mod 4 = 0" does not mean the first term of the cycle. When the reduced exponent is 0 (as with $7^{2024}$ ), the answer is the fourth (last) term of the cycle, not the first — landing on $7$ instead of $1$ is the classic slip. Same family: a units-digit cycle answers units digit questions only; a remainder by 7 or 11 needs the full modular reduction, not the digit cycle.

Watch this

2IIM's number-theory blitzkrieg — every CAT number-system question across years, solved:

🎯PYQ Evidence

Number system runs on three engines: remainder cycles, prime factorisation, and place value. : powers of 10 mod 7 repeat with cycle length 6 (10⁶ ≡ 1), so 10¹⁰⁰ ≡ 10⁴ because 100 = 6×16 + 4, giving remainder 4. : factor 168 = 2³·3·7 and 1134 = 2·3⁴·7, then "A divides B" just means every prime's power in A is ≤ that in B, which fixes the least n and m prime by prime. : writing the number as 100c+10b+a, the reverse-minus-original cancels the tens digit and leaves 99(a−c) = 198, so a−c = 2 and you just count the free choices (70). Decide which engine the question wants, and the rest is bookkeeping.

Checklist

Prime-factorise first; then (a+1)(b+1)… for factor counts
Use HCF × LCM = product to find the missing one
For powers, take the exponent mod 4 (units digit) or mod (p−1) (Fermat)
Exponent ≡ 0 → the last term of the cycle
Always reduce to residues before multiplying or powering
Know the divisibility tests for 3, 4, 8, 9, 11 cold

Sample Questions

42 practice questions

Hard

If q, r, and s are consecutive even integers and q < r < s, which of the following CANNOT be the value of $s^{2}$ - $r^{2}$ - $q^{2}$ ?

Hard

If x represents the sum of all positive three-digit numbers formed using each of the distinct nonzero digits a, b, and c exactly once, what is the largest integer by which x must be divisible?

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CAT PYQ Spotlight

Actual CAT questions on this topic

CAT 2025 · Slot 1

TITAHard

In a 3-digit number N, the digits are non-zero and distinct such that none of the digits is a perfect square, and only one of the digits is a prime number. Then, the number of factors of the minimum possible value of N is

Your answer

CAT 2024 · Slot 1

Easy

When $10^{100}$ is divided by 7, the remainder is

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