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Lesson 16 min· Section 1 of 4

The straight-up lesson, built for the exam.

Factors & Multiples

Formula sheet

Everything for this chapter, in one place. Throughout, $N = p^a q^b r^c \cdots$ is the prime factorisation.

Counting divisors


Number of divisors	$(a+1)(b+1)(c+1)\cdots$
Sum of divisors	$\dfrac{p^{a+1}-1}{p-1} \times \dfrac{q^{b+1}-1}{q-1} \times \cdots$
Product of divisors	$N^{\,d/2}$ , where $d$ is the number of divisors
Perfect-square divisors	count only the even exponents: $\left(\lfloor\frac{a}{2}\rfloor+1\right)\left(\lfloor\frac{b}{2}\rfloor+1\right)\cdots$
Perfect-cube divisors	the same with multiples of 3

What a divisor count tells you


Exactly 2 divisors	a prime
Exactly 3 divisors	$p^2$
Exactly 4 divisors	$p^3$ or $pq$
Exactly $d$ divisors	every way of writing $d$ as a product gives an exponent pattern
$d$ is odd	exactly when $N$ is a perfect square

Factor pairs


Pairs $(a,b)$ with $ab = N$ and $a \le b$	$\left\lceil \dfrac{d}{2} \right\rceil$
$x^2 - y^2 = N$	factor $N$ as $(x-y)(x+y)$ ; both factors must share parity
$\dfrac{1}{a}+\dfrac{1}{b} = \dfrac{1}{k}$	rearranges to $(a-k)(b-k) = k^2$
Any "product equals a constant"	list the factor pairs, including negative ones when integers are allowed

Divisibility


If $k$ divides $A$ and $B$	$k$ divides every $mA + nB$
Count up to $n$ divisible by $a$	$\left\lfloor \dfrac{n}{a} \right\rfloor$
Divisible by none of $a,b,c$	inclusion–exclusion on the counts
Highest power of a prime $p$ in $n!$	$\left\lfloor\frac{n}{p}\right\rfloor + \left\lfloor\frac{n}{p^2}\right\rfloor + \cdots$

Matching exponents


$m^n = p^a q^b$	$n$ must divide every exponent, so $n \mid \gcd(a,b)$
Smallest $n$ with $A \mid B^n$	take the worst prime: $n = \max\lceil \frac{\text{needed}}{\text{available}} \rceil$
Largest integer dividing a family	test the first two or three members and take their gcd

Why this chapter matters

0.8

questions in a typical Quant section

17th

of 25 chapters

≈ 2 of 66 marks a section ride on it.

Asked in12 of 24 papers

20172025

Year by yearper section

2017: 0.72017: 0.7 per section
2018: 0.72018: 0.7 per section
2019: 12019: 1 per section
2020: 0.82020: 0.8 per section
2021: 02021: 0 per section
2022: 12022: 1 per section
2023: 1.72023: 1.7 per section
2024: 0.32024: 0.3 per section
2025: 12025: 1 per section

Question type22 asked

MCQ 54%Type-in 46%

Type-in answers carry no negative marking.

From every Factors & Multiples question in CAT papers since 2017.