🔢 Quant

Divisibility & Remainders

Factors, divisibility rules, remainders, and cyclicity.

of Quant

Why This Topic Matters

Total PYQs📊

of 1002 · 2021–2025

Years featured📅

4/5

of recent CAT years

% of Quant📈

~5%

of section questions

Est. hours⏱️

~6h

to master

2021

~2/22

2022

~2/22

2023

~2/22

2024

~2/22

2025

🎯PYQ Evidence

CAT 2021–2025: ~1.2 per slot (2022: 1.7 · 2023: 1.7 · 2024: 1.3 · 2025: 1.3). Remainder questions and factor/LCM–HCF questions alternate — together about 1 per slot (2021's number-system load sat in digits/properties instead).

Factors, Divisibility, HCF & LCM

Everything here flows from one move: prime-factorise first. Once a number is written as $p^a q^b r^c$ , factor counts, divisor sums, HCF and LCM all follow mechanically.

Core results

For $N=p^{a}q^{b}r^{c}$ :

Number of factors $=(a+1)(b+1)(c+1)$ .
Sum of factors $=\dfrac{p^{a+1}-1}{p-1}\cdot\dfrac{q^{b+1}-1}{q-1}\cdot\dfrac{r^{c+1}-1}{r-1}$ .
HCF takes the lowest power of each common prime; LCM the highest.
$\text{HCF}\times\text{LCM}=$ product of the two numbers.

Divisibility tests

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

A worked example

How many factors does 720 have, and how many are even?

Prime-factorise: $720=2^{4}\cdot3^{2}\cdot5^{1}$ . Total factors:

$(4+1)(2+1)(1+1)=5\cdot3\cdot2=\mathbf{30}.$

Even factors must contain at least one 2: fix the power of 2 from $\{1,2,3,4\}$ (4 choices) and combine freely with $3^{0..2},5^{0..1}$ :

$4\cdot3\cdot2=\mathbf{24}\ \text{even factors}\ (\text{so }30-24=6\text{ are odd}).$

🎯PYQ Evidence

Prime exponents and remainder cycles do all the heavy lifting. : powers of 10 cycle mod 7 with length 6 (since 10^6 ≡ 1), so 100 = 6×16 + 4 lands on 10^4 ≡ 4. : a divides b exactly when every prime's exponent in a ≤ that in b, so factor 168 = 2^3·3·7 and 1134 = 2·3^4·7, match powers to get n = 3 and m = 12, giving m + n = 15. : a divisor 2^a·3^b·5^c·7^d is ≡ 1 mod 3 only if b = 0, then 2,5 ≡ −1 force a + c even — 14 same-parity pairs × 3 free choices of d = 42. Whether it's remainders, divisibility, or counting divisors, reduce to prime exponents or a short remainder cycle.

Common traps

Counting 1 and $N$ . Both are factors — don't drop them.
Perfect squares. They have an odd number of factors (one factor is unpaired — the square root).
HCF vs LCM mix-up. HCF = lowest powers (small), LCM = highest powers (large).

Checklist

Prime-factorise before anything else
Factor count $=(a+1)(b+1)\dots$
Even factors: force one prime to appear at power $\ge 1$
$\text{HCF}\times\text{LCM}=$ product of the numbers

Sample Questions

64 practice questions

Easy

What is the units digit of $17^{27}$ ?

Medium

What is the remainder when ( $1^{1}$ + $2^{2}$ + $3^{3}$ + $4^{4}$ + $5^{5}$ + $6^{6}$ + $7^{7}$ + $8^{8}$ + $9^{9}$ + $10^{10}$ ) is divided by 5?

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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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