Class 10 Maths Formula Sheet

Class 10 is a pivotal year for students, with board exams setting the stage for future academic pursuits. Mathematics is a core subject where scoring highly can significantly boost your overall percentage. This Class 10 Maths formula sheet is meticulously designed to help you achieve exactly that. It covers all the fundamental formulas from the syllabus, including algebra, geometry, trigonometry, mensuration, probability, and number theory. Having a consolidated list of formulas makes revision much more efficient, allowing you to quickly review key concepts before an exam. Practice applying these formulas to various types of problems to ensure you not only remember them but also understand their practical application. This resource is perfect for daily review and is guaranteed to enhance your problem-solving speed and accuracy during the actual board exams.

algebra

Quadratic Formula

Class 10algebra

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Roots of a quadratic equation ax^2 + bx + c = 0

#roots#equations#quadratic

Difference of Squares

Class 10algebra

a^2 - b^2 = (a - b)(a + b)

Factoring difference of two squares

#factoring#polynomials

Sum of Cubes

Class 10algebra

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

Factoring sum of two cubes

#factoring#cubes

Difference of Cubes

Class 10algebra

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

Factoring difference of two cubes

#factoring#cubes

Binomial Theorem

Class 12algebra

(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k

Expansion of a binomial power

#expansion#series#binomial

Arithmetic Progression (nth Term)

Class 10algebra

a_n = a_1 + (n-1)d

nth term of an AP with common difference d

#sequences#ap

Arithmetic Progression (Sum)

Class 10algebra

S_n = \frac{n}{2}(2a_1 + (n-1)d) = \frac{n}{2}(a_1 + a_n)

Sum of first n terms of an AP

#series#ap

Geometric Progression (nth Term)

Class 12algebra

a_n = a_1 r^{n-1}

nth term of a GP with common ratio r

#sequences#gp

Geometric Progression (Sum)

Class 12algebra

S_n = \frac{a_1(1-r^n)}{1-r} \quad (r \neq 1)

Sum of first n terms of a GP

#series#gp

Infinite Geometric Series

Class 12algebra

S_\infty = \frac{a_1}{1-r} \quad (|r| < 1)

Sum of infinite GP when ratio is strictly between -1 and 1

#series#infinite

Vieta's Formulas (Quadratic)

Class 10algebra

x_1 + x_2 = -\frac{b}{a}, \quad x_1 x_2 = \frac{c}{a}

Sum and product of roots of ax^2 + bx + c = 0

#roots#vieta#polynomials

Completing the Square

Class 10algebra

ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right)

Rewriting quadratic into vertex form

#quadratic#factoring

geometry

Distance Formula (2D)

Class 10geometry

d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

Distance between two points (x1, y1) and (x2, y2)

Interactive Solver:

Point 1 (x₁, y₁)

Point 2 (x₂, y₂)

Distance:5

Midpoint:(1.5, 2)

Slope (m):1.333

#coordinate#distance

Midpoint Formula

Class 10geometry

M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)

Midpoint of a line segment

#coordinate#midpoint

Slope of a Line

Class 10geometry

m = \frac{y_2-y_1}{x_2-x_1} = \tan\theta

Slope of line passing through two points

#coordinate#line#slope

Slope-Intercept Form

Class 10geometry

y = mx + c

Equation of a straight line with slope m and y-intercept c

#coordinate#line

Point-Slope Form

Class 12geometry

y - y_1 = m(x - x_1)

Equation of line passing through (x1, y1) with slope m

#coordinate#line

Standard Circle Equation

Class 12geometry

(x-h)^2 + (y-k)^2 = r^2

Circle with center (h, k) and radius r

#conic#circle

General Parabola Equation

Class 12geometry

y = a(x-h)^2 + k \quad \text{or} \quad (y-k)^2 = 4a(x-h)

Parabola in vertex form

#conic#parabola

Standard Ellipse Equation

Class 12geometry

\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1

Ellipse centered at (h, k) with semi-major/minor axes a, b

#conic#ellipse

Standard Hyperbola Equation

Class 12geometry

\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1

Horizontal hyperbola centered at (h, k)

#conic#hyperbola

Section Formula (Internal)

Class 10geometry

P = \left(\frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n}\right)

Coordinates of point dividing line segment in ratio m:n

Interactive Solver:

Point A (x₁, y₁)

Point B (x₂, y₂)

Ratio m : n

Point P:(2, 4)

#coordinate#section

Perpendicular Distance from Point to Line

Class 12geometry

d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}

Distance from point (x1, y1) to line Ax + By + C = 0

#coordinate#distance#line

trigonometry

Pythagorean Identity

Class 10trigonometry

\sin^2\theta + \cos^2\theta = 1

Fundamental trigonometric identity relating sine and cosine

#identities#pythagorean

Secant & Tangent Identity

Class 10trigonometry

1 + \tan^2\theta = \sec^2\theta

Pythagorean identity for tangent and secant

#identities#pythagorean

Cosecant & Cotangent Identity

Class 10trigonometry

1 + \cot^2\theta = \csc^2\theta

Pythagorean identity for cotangent and cosecant

#identities#pythagorean

Sine Addition & Subtraction

Class 12trigonometry

\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B

Sine of sum or difference of two angles

#compound#addition

Cosine Addition & Subtraction

Class 12trigonometry

\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B

Cosine of sum or difference of two angles

#compound#addition

Tangent Addition & Subtraction

Class 12trigonometry

\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}

Tangent of sum or difference of two angles

#compound#addition

Double Angle (Sine)

Class 12trigonometry

\sin(2\theta) = 2\sin\theta\cos\theta

Sine of double angle

#double-angle

Double Angle (Cosine)

Class 12trigonometry

\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta

Cosine of double angle formulas

#double-angle

Half Angle Formulas

Class 12trigonometry

\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1-\cos\theta}{2}}, \quad \cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1+\cos\theta}{2}}

Sine and cosine of half angle

#half-angle

Law of Sines

Class 10trigonometry

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R

Relates sides of a triangle to the sines of its angles

Interactive Triangle Solver (SAS):

Side a

Side b

Angle C (°)

Computed Side c:c = 5.568

#triangles#sines

Law of Cosines

Class 10trigonometry

c^2 = a^2 + b^2 - 2ab\cos C

Relates sides of a triangle to cosine of an angle

Interactive Triangle Solver (SAS):

Side a

Side b

Angle C (°)

Computed Side c:c = 5.568

#triangles#cosines

mensuration

Circle Area

mensuration

A = \pi r^2

Area of a circle with radius r

#2d#area#circle

Circle Circumference

mensuration

C = 2\pi r

Perimeter of a circle

#2d#perimeter#circle

Triangle Area

mensuration

A = \frac{1}{2}bh

Area of a triangle with base b and height h

#2d#area#triangle

Heron's Formula

mensuration

A = \sqrt{s(s-a)(s-b)(s-c)}

Triangle area from side lengths, s = (a+b+c)/2

#2d#area#triangle

Rectangle Area

mensuration

A = l \times w

Area of a rectangle

#2d#area#rectangle

Trapezium Area

mensuration

A = \frac{1}{2}(a+b)h

Area of a trapezium with parallel sides a, b

#2d#area#trapezium

Sphere Volume

mensuration

V = \frac{4}{3}\pi r^3

Volume of a sphere

#3d#volume#sphere

Sphere Surface Area

mensuration

S = 4\pi r^2

Surface area of a sphere

#3d#surface#sphere

Cylinder Volume

mensuration

V = \pi r^2 h

Volume of a cylinder

#3d#volume#cylinder

Cylinder Surface Area

mensuration

S = 2\pi r(r+h)

Total surface area of a cylinder

#3d#surface#cylinder

Cone Volume

mensuration

V = \frac{1}{3}\pi r^2 h

Volume of a cone

#3d#volume#cone

Cone Curved Surface Area

mensuration

S = \pi r l

Lateral surface area of a cone with slant height l

#3d#surface#cone

Cuboid Volume

mensuration

V = l \times w \times h

Volume of a cuboid

#3d#volume#cuboid

Hemisphere Volume

mensuration

V = \frac{2}{3}\pi r^3

Volume of a hemisphere

#3d#volume#hemisphere

probability

Permutations Formula

probability

P(n, r) = {}^n P_r = \frac{n!}{(n-r)!}

Number of ordered arrangements of r elements from n

#combinatorics#permutations

Combinations Formula

probability

C(n, r) = {}^n C_r = \binom{n}{r} = \frac{n!}{r!(n-r)!}

Number of unordered selections of r elements from n

#combinatorics#combinations

Addition Rule of Probability

probability

P(A \cup B) = P(A) + P(B) - P(A \cap B)

Probability of union of two events

#probability#union

Multiplication Rule

probability

P(A \cap B) = P(A) \cdot P(B|A)

Probability of simultaneous occurrence of two events

#probability#intersection

Conditional Probability

probability

P(A|B) = \frac{P(A \cap B)}{P(B)} \quad (P(B) > 0)

Probability of event A occurring given event B has occurred

#probability#conditional

Bayes' Theorem

probability

P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} = \frac{P(B|A) P(A)}{\sum P(B|A_i) P(A_i)}

Calculates posterior probability based on prior knowledge

#probability#bayes

Binomial Distribution PMF

probability

P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}

Probability of k successes in n independent Bernoulli trials

#distribution#binomial

Poisson Distribution PMF

probability

P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}

Probability of k events in a fixed interval with average rate λ

#distribution#poisson

Expected Value (Discrete)

probability

E[X] = \sum_{i} x_i \cdot P(X = x_i)

Weighted average of all possible values of discrete random variable

#statistics#mean#expectation

Variance & Standard Deviation

probability

\text{Var}(X) = E[X^2] - (E[X])^2, \quad \sigma = \sqrt{\text{Var}(X)}

Measure of dispersion of a random variable

#statistics#variance

Standard Normal Distribution (PDF)

probability

f(z) = \frac{1}{\sqrt{2\pi}} e^{-z^2/2}, \quad z = \frac{x - \mu}{\sigma}

Probability density function of standard normal variable

#distribution#normal#z-score

number Theory

GCD-LCM Relationship

numberTheory

\gcd(a,b) \times \text{lcm}(a,b) = |ab|

Product of GCD and LCM equals product of numbers

#gcd#lcm

Fermat's Little Theorem

numberTheory

a^{p-1} \equiv 1 \pmod{p}

For prime p and integer a not divisible by p

#modular#prime#fermat

Euler's Totient Theorem

numberTheory

a^{\phi(n)} \equiv 1 \pmod{n}

Generalization of Fermat's Little Theorem

#modular#euler#totient

Wilson's Theorem

numberTheory

(p-1)! \equiv -1 \pmod{p}

A number p is prime iff this holds

#prime#factorial#wilson

Euclidean Algorithm

numberTheory

\gcd(a,b) = \gcd(b, a \bmod b)

Recursive algorithm for finding GCD

#gcd#algorithm#euclidean

Divisibility Rule for 3

numberTheory

n \equiv 0 \pmod{3} \iff \text{digit sum} \equiv 0 \pmod{3}

A number is divisible by 3 if its digit sum is

#divisibility#rules

Modular Addition

numberTheory

(a + b) \bmod n = ((a \bmod n) + (b \bmod n)) \bmod n

Modular arithmetic addition property

#modular#addition

Modular Multiplication

numberTheory

(a \cdot b) \bmod n = ((a \bmod n) \cdot (b \bmod n)) \bmod n

Modular arithmetic multiplication property

#modular#multiplication

Chinese Remainder Theorem

numberTheory

x \equiv a_i \pmod{n_i} \text{ has unique solution mod } N = \prod n_i

System of congruences with coprime moduli

#crt#congruence

Prime Counting (Approx)

numberTheory

\pi(x) \sim \frac{x}{\ln x}

Approximate number of primes up to x

#prime#counting#approximation

Related Formula Sheets

Class 12 Maths Formula Sheet GCSE Maths Formula Sheet IGCSE Maths Formula Sheet

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Class 10 Maths Formula Sheet

algebra

Quadratic Formula

Class 10algebra

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Roots of a quadratic equation ax^2 + bx + c = 0

#roots#equations#quadratic

Difference of Squares

Class 10algebra

a^2 - b^2 = (a - b)(a + b)

Factoring difference of two squares

#factoring#polynomials

Sum of Cubes

Class 10algebra

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

Factoring sum of two cubes

#factoring#cubes

Difference of Cubes

Class 10algebra

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

Factoring difference of two cubes

#factoring#cubes

Binomial Theorem

Class 12algebra

(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k

Expansion of a binomial power

#expansion#series#binomial

Arithmetic Progression (nth Term)

Class 10algebra

a_n = a_1 + (n-1)d

nth term of an AP with common difference d

#sequences#ap

Arithmetic Progression (Sum)

Class 10algebra

S_n = \frac{n}{2}(2a_1 + (n-1)d) = \frac{n}{2}(a_1 + a_n)

Sum of first n terms of an AP

#series#ap

Geometric Progression (nth Term)

Class 12algebra

a_n = a_1 r^{n-1}

nth term of a GP with common ratio r

#sequences#gp

Geometric Progression (Sum)

Class 12algebra

S_n = \frac{a_1(1-r^n)}{1-r} \quad (r \neq 1)

Sum of first n terms of a GP

#series#gp

Infinite Geometric Series

Class 12algebra

S_\infty = \frac{a_1}{1-r} \quad (|r| < 1)

Sum of infinite GP when ratio is strictly between -1 and 1

#series#infinite

Vieta's Formulas (Quadratic)

Class 10algebra

x_1 + x_2 = -\frac{b}{a}, \quad x_1 x_2 = \frac{c}{a}

Sum and product of roots of ax^2 + bx + c = 0

#roots#vieta#polynomials

Completing the Square

Class 10algebra

ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right)

Rewriting quadratic into vertex form

#quadratic#factoring

geometry

Distance Formula (2D)

Class 10geometry

d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

Distance between two points (x1, y1) and (x2, y2)

Interactive Solver:

Point 1 (x₁, y₁)

Point 2 (x₂, y₂)

Distance:5

Midpoint:(1.5, 2)

Slope (m):1.333

#coordinate#distance

Midpoint Formula

Class 10geometry

M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)

Midpoint of a line segment

#coordinate#midpoint

Slope of a Line

Class 10geometry

m = \frac{y_2-y_1}{x_2-x_1} = \tan\theta

Slope of line passing through two points

#coordinate#line#slope

Slope-Intercept Form

Class 10geometry

y = mx + c

Equation of a straight line with slope m and y-intercept c

#coordinate#line

Point-Slope Form

Class 12geometry

y - y_1 = m(x - x_1)

Equation of line passing through (x1, y1) with slope m

#coordinate#line

Standard Circle Equation

Class 12geometry

(x-h)^2 + (y-k)^2 = r^2

Circle with center (h, k) and radius r

#conic#circle

General Parabola Equation

Class 12geometry

y = a(x-h)^2 + k \quad \text{or} \quad (y-k)^2 = 4a(x-h)

Parabola in vertex form

#conic#parabola

Standard Ellipse Equation

Class 12geometry

\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1

Ellipse centered at (h, k) with semi-major/minor axes a, b

#conic#ellipse

Standard Hyperbola Equation

Class 12geometry

\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1

Horizontal hyperbola centered at (h, k)

#conic#hyperbola

Section Formula (Internal)

Class 10geometry

P = \left(\frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n}\right)

Coordinates of point dividing line segment in ratio m:n

Interactive Solver:

Point A (x₁, y₁)

Point B (x₂, y₂)

Ratio m : n

Point P:(2, 4)

#coordinate#section

Perpendicular Distance from Point to Line

Class 12geometry

d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}

Distance from point (x1, y1) to line Ax + By + C = 0

#coordinate#distance#line

trigonometry

Pythagorean Identity

Class 10trigonometry

\sin^2\theta + \cos^2\theta = 1

Fundamental trigonometric identity relating sine and cosine

#identities#pythagorean

Secant & Tangent Identity

Class 10trigonometry

1 + \tan^2\theta = \sec^2\theta

Pythagorean identity for tangent and secant

#identities#pythagorean

Cosecant & Cotangent Identity

Class 10trigonometry

1 + \cot^2\theta = \csc^2\theta

Pythagorean identity for cotangent and cosecant

#identities#pythagorean

Sine Addition & Subtraction

Class 12trigonometry

\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B

Sine of sum or difference of two angles

#compound#addition

Cosine Addition & Subtraction

Class 12trigonometry

\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B

Cosine of sum or difference of two angles

#compound#addition

Tangent Addition & Subtraction

Class 12trigonometry

\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}

Tangent of sum or difference of two angles

#compound#addition

Double Angle (Sine)

Class 12trigonometry

\sin(2\theta) = 2\sin\theta\cos\theta

Sine of double angle

#double-angle

Double Angle (Cosine)

Class 12trigonometry

\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta

Cosine of double angle formulas

#double-angle

Half Angle Formulas

Class 12trigonometry

\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1-\cos\theta}{2}}, \quad \cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1+\cos\theta}{2}}

Sine and cosine of half angle

#half-angle

Law of Sines

Class 10trigonometry

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R

Relates sides of a triangle to the sines of its angles

Interactive Triangle Solver (SAS):

Side a

Side b

Angle C (°)

Computed Side c:c = 5.568

#triangles#sines

Law of Cosines

Class 10trigonometry

c^2 = a^2 + b^2 - 2ab\cos C

Relates sides of a triangle to cosine of an angle

Interactive Triangle Solver (SAS):

Side a

Side b

Angle C (°)

Computed Side c:c = 5.568

#triangles#cosines

mensuration

Circle Area

mensuration

A = \pi r^2

Area of a circle with radius r

#2d#area#circle

Circle Circumference

mensuration

C = 2\pi r

Perimeter of a circle

#2d#perimeter#circle

Triangle Area

mensuration

A = \frac{1}{2}bh

Area of a triangle with base b and height h

#2d#area#triangle

Heron's Formula

mensuration

A = \sqrt{s(s-a)(s-b)(s-c)}

Triangle area from side lengths, s = (a+b+c)/2

#2d#area#triangle

Rectangle Area

mensuration

A = l \times w

Area of a rectangle

#2d#area#rectangle

Trapezium Area

mensuration

A = \frac{1}{2}(a+b)h

Area of a trapezium with parallel sides a, b

#2d#area#trapezium

Sphere Volume

mensuration

V = \frac{4}{3}\pi r^3

Volume of a sphere

#3d#volume#sphere

Sphere Surface Area

mensuration

S = 4\pi r^2

Surface area of a sphere

#3d#surface#sphere

Cylinder Volume

mensuration

V = \pi r^2 h

Volume of a cylinder

#3d#volume#cylinder

Cylinder Surface Area

mensuration

S = 2\pi r(r+h)

Total surface area of a cylinder

#3d#surface#cylinder

Cone Volume

mensuration

V = \frac{1}{3}\pi r^2 h

Volume of a cone

#3d#volume#cone

Cone Curved Surface Area

mensuration

S = \pi r l

Lateral surface area of a cone with slant height l

#3d#surface#cone

Cuboid Volume

mensuration

V = l \times w \times h

Volume of a cuboid

#3d#volume#cuboid

Hemisphere Volume

mensuration

V = \frac{2}{3}\pi r^3

Volume of a hemisphere

#3d#volume#hemisphere

probability

Permutations Formula

probability

P(n, r) = {}^n P_r = \frac{n!}{(n-r)!}

Number of ordered arrangements of r elements from n

#combinatorics#permutations

Combinations Formula

probability

C(n, r) = {}^n C_r = \binom{n}{r} = \frac{n!}{r!(n-r)!}

Number of unordered selections of r elements from n

#combinatorics#combinations

Addition Rule of Probability

probability

P(A \cup B) = P(A) + P(B) - P(A \cap B)

Probability of union of two events

#probability#union

Multiplication Rule

probability

P(A \cap B) = P(A) \cdot P(B|A)

Probability of simultaneous occurrence of two events

#probability#intersection

Conditional Probability

probability

P(A|B) = \frac{P(A \cap B)}{P(B)} \quad (P(B) > 0)

Probability of event A occurring given event B has occurred

#probability#conditional

Bayes' Theorem

probability

P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} = \frac{P(B|A) P(A)}{\sum P(B|A_i) P(A_i)}

Calculates posterior probability based on prior knowledge

#probability#bayes

Binomial Distribution PMF

probability

P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}

Probability of k successes in n independent Bernoulli trials

#distribution#binomial

Poisson Distribution PMF

probability

P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}

Probability of k events in a fixed interval with average rate λ

#distribution#poisson

Expected Value (Discrete)

probability

E[X] = \sum_{i} x_i \cdot P(X = x_i)

Weighted average of all possible values of discrete random variable

#statistics#mean#expectation

Variance & Standard Deviation

probability

\text{Var}(X) = E[X^2] - (E[X])^2, \quad \sigma = \sqrt{\text{Var}(X)}

Measure of dispersion of a random variable

#statistics#variance

Standard Normal Distribution (PDF)

probability

f(z) = \frac{1}{\sqrt{2\pi}} e^{-z^2/2}, \quad z = \frac{x - \mu}{\sigma}

Probability density function of standard normal variable

#distribution#normal#z-score

number Theory

GCD-LCM Relationship

numberTheory

\gcd(a,b) \times \text{lcm}(a,b) = |ab|

Product of GCD and LCM equals product of numbers

#gcd#lcm

Fermat's Little Theorem

numberTheory

a^{p-1} \equiv 1 \pmod{p}

For prime p and integer a not divisible by p

#modular#prime#fermat

Euler's Totient Theorem

numberTheory

a^{\phi(n)} \equiv 1 \pmod{n}

Generalization of Fermat's Little Theorem

#modular#euler#totient

Wilson's Theorem

numberTheory

(p-1)! \equiv -1 \pmod{p}

A number p is prime iff this holds

#prime#factorial#wilson

Euclidean Algorithm

numberTheory

\gcd(a,b) = \gcd(b, a \bmod b)

Recursive algorithm for finding GCD

#gcd#algorithm#euclidean

Divisibility Rule for 3

numberTheory

n \equiv 0 \pmod{3} \iff \text{digit sum} \equiv 0 \pmod{3}

A number is divisible by 3 if its digit sum is

#divisibility#rules

Modular Addition

numberTheory

(a + b) \bmod n = ((a \bmod n) + (b \bmod n)) \bmod n

Modular arithmetic addition property

#modular#addition

Modular Multiplication

numberTheory

(a \cdot b) \bmod n = ((a \bmod n) \cdot (b \bmod n)) \bmod n

Modular arithmetic multiplication property

#modular#multiplication

Chinese Remainder Theorem

numberTheory

x \equiv a_i \pmod{n_i} \text{ has unique solution mod } N = \prod n_i

System of congruences with coprime moduli

#crt#congruence

Prime Counting (Approx)

numberTheory

\pi(x) \sim \frac{x}{\ln x}

Approximate number of primes up to x

#prime#counting#approximation