How many Remainders & Divisibility questions come in CAT Quant?

About 0.3 questions in a 22-question Quant section, worth roughly 1 marks. Remainders & Divisibility appeared in 4 of the 24 CAT papers from 2017 to 2025, 4 questions in all, which ranks it 23 of 25 Quant chapters.

Are Remainders & Divisibility questions in CAT MCQ or TITA?

25% of the Remainders & Divisibility questions asked since 2017 were TITA (type in the answer, no negative marking). The rest were multiple choice.

What types of Remainders & Divisibility questions does CAT ask?

CAT keeps returning to 2 question shapes in Remainders & Divisibility: The remainder of a large power; The same remainder from several divisors.

Remainders & Divisibility for CAT: Formulas, Weightage & Question Types

Remainders of large powers, and the same remainder from several divisors.

Remainders & Divisibility formula sheet

Everything CAT needs from Remainders & Divisibility, in one place. Bookmark it for revision; the lesson in the app explains where each line comes from and when to reach for it.

Everything for this chapter, in one place.

The smallest chapter that gets asked — 4 questions in nine years. But all four are recent and all four are in the current format, so unlike Mensuration this one is on the way up, not down.

Working with remainders


Notation	$a \equiv b \pmod m$ means $a$ and $b$ leave the same remainder
Addition	$(a+b) \bmod m = \bigl((a \bmod m) + (b \bmod m)\bigr) \bmod m$
Multiplication	the same — replace each factor by its remainder at any point
So for powers	replace the base by its remainder first, then work on the exponent

Powers cycle


The remainders of $a^1, a^2, a^3, \dots$	repeat with some cycle length $c$
So only the exponent mod $c$ matters	$a^n \equiv a^{\,n \bmod c}$ , taking $c$ when the remainder is 0
Fermat, for prime $p$ not dividing $a$	$a^{p-1} \equiv 1 \pmod p$ , so $c$ divides $p-1$
Finding $c$	just list powers until you hit 1

The negative-remainder shortcut


$10 \equiv -1 \pmod{11}$	so $10^{\text{even}} \equiv 1$ and $10^{\text{odd}} \equiv -1$
In general, if $a \equiv -1$	the cycle is only 2 long
If $a \equiv 1$	every power is 1
A remainder of $-k$	means a true remainder of $m-k$

A common remainder


Same remainder $r$ for divisors $d_1, d_2, \dots$	$N = \operatorname{lcm}(d_1,d_2,\dots)\,k + r$
Add a further divisibility condition	substitute and solve for $k$ as a congruence
Bounded $N$	bounds $k$ , usually to a handful of cases

Divisibility tests


3 or 9	digit sum
11	alternating digit sum
4 / 8	last two / last three digits
7, 11, 13 together	they divide $1001$ , so grouping digits in threes works

The 2 Remainders & Divisibility question types CAT repeats

Past CAT papers return to the same few shapes. Learn to recognise them and most questions become a known routine.

01
The remainder of a large power
02
The same remainder from several divisors

Is Remainders & Divisibility costing you marks?

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Remainders & Divisibility in CAT: quick answers

How many Remainders & Divisibility questions come in CAT Quant?: About 0.3 questions in a 22-question Quant section, worth roughly 1 marks. Remainders & Divisibility appeared in 4 of the 24 CAT papers from 2017 to 2025, 4 questions in all, which ranks it 23 of 25 Quant chapters.
Are Remainders & Divisibility questions in CAT MCQ or TITA?: 25% of the Remainders & Divisibility questions asked since 2017 were TITA (type in the answer, no negative marking). The rest were multiple choice.
What types of Remainders & Divisibility questions does CAT ask?: CAT keeps returning to 2 question shapes in Remainders & Divisibility: The remainder of a large power; The same remainder from several divisors.

Remainders & Divisibility for CAT: Formulas, Weightage & Question Types

Remainders & Divisibility formula sheet

The 2 Remainders & Divisibility question types CAT repeats

Is Remainders & Divisibility costing you marks?

Remainders & Divisibility in CAT: quick answers

More Number System for CAT