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CAT Quantitative Aptitude Formulas & Shortcuts

Essential formulas in Number Systems, Geometry, Algebra, and Arithmetic for CAT aspirants.

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Euler's Totient Function

ϕ (N)

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Euler's Totient Function

ϕ (N)

N = p_{1}^{a} p_{2}^{b}

, then $

ϕ (N) = N (1 - \frac{1}{p _{1}}) (1 - \frac{1}{p _{2}})

Euler's Theorem for Remainders

g cd (a, m) = 1

, then $

a^{ϕ (m)} \equiv 1 (mod m)

Fermat's Little Theorem

p

is prime and

g cd (a, p) = 1

, then $

a^{p - 1} \equiv 1 (mod p)

Wilson's Theorem

For any prime

p

: $

(p - 1)! \equiv - 1 (mod p)

Apollonius' Theorem (Median of Triangle)

For median

A D

to side

B C

: $

A B^{2} + A C^{2} = 2 (A D^{2} + B D^{2})

Stewart's Theorem (Cevian length)

For cevian

d

dividing

a

into segments

m

and

n

: $

b^{2} m + c^{2} n = a (d^{2} + mn)

Inradius of a Triangle

r = \frac{Δ}{s}

w h er e

\Delta$ is area and $s = \frac{a+b+c}{2}$ is semiperimeter.

Circumradius of a Triangle

R = \frac{ab c}{4Δ}

Ptolemy's Theorem for Cyclic Quadrilaterals

Product of diagonals equals sum of products of opposite sides: $

A C \cdot B D = A B \cdot C D + B C \cdot A D

Brahmagupta's Formula for Cyclic Quadrilaterals

Δ = (s - a) (s - b) (s - c) (s - d)

Number of Integral Solutions to

x_{1} + x_{2} + \dots + x_{r} = n

(non-negative)

(r - 1 n + r - 1)

AM-GM-HM Inequality

AM \geq GM \geq HM

$ Equality holds if and only if all terms are equal.

Alligation Rule for Mixtures

\frac{Quantity of Cheaper}{Quantity of Dearer} = \frac{Price of Dearer - Mean Price}{Mean Price - Price of Cheaper}

Replacement in Mixtures Formula

After

n

operations of replacing

y

units from

x

units: $

Remaining = x (1 - \frac{y}{x})^{n}

Sum of First

n

Squares

\sum k^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}

Sum of First

n

Cubes

\sum k^{3} = [\frac{n ( n + 1 )}{2}]^{2}

Number of Trailing Zeros in

n!

k = 1 \sum \infty ⌊ \frac{n}{5 ^{k}} ⌋ = ⌊ \frac{n}{5} ⌋ + ⌊ \frac{n}{25} ⌋ + \dots

Clocks: Angle Between Hands at

H

hours and

M

minutes

θ = 30 H - \frac{11}{2} M

Circular Tracks: Number of Meeting Points

Opposite directions:

a + b

distinct points. Same direction:

∣ a - b ∣

distinct points, where speed ratio is reduced to

a : b

Escalator Problems Relationship

Total Steps = T \times (v_{person} \pm v_{escalator})

Logarithm Base Change Formula

lo g_{b} (a) = \frac{lo g _{c} ( a )}{lo g _{c} ( b )} = \frac{1}{lo g _{a} ( b )}

Descartes' Rule of Signs

The number of positive real roots of

P (x)

is at most the number of sign changes in coefficients, or less by an even integer.

Area of Equilateral Triangle

A = \frac{3}{4} s^{2}

Area of Hexagon of side

s

A = 6 \times \frac{3}{4} s^{2} = \frac{3 3}{2} s^{2}

Difference of Cubes Factorization

a^{3} - b^{3} = (a - b) (a^{2} + ab + b^{2})

CAT Quantitative Aptitude Formulas & Shortcuts

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