Math formulas

Each bit is a coefficient of a power of 2.

Enter a positive integer n within the educational size limit. Related: Toy RSA Demonstrator.

General real powers use the exponential definition.

Enter a and modulus m. If gcd ≠ 1, no modular inverse exists (shown as —). Related: Modular Exponentiation, Toy RSA Demonstrator.

When an inverse exists, a·inv ≡ 1 (mod m).

Simplify by dividing by the greatest common divisor.

Addition uses a common denominator.

Enter Integer a and Integer b for the GCD and LCM Calculator; use finite values in the indicated domain. The calculator reports GCD and LCM: GCD is the largest shared divisor; LCM is the least common multiple. When either integer is zero, check the LCM convention required by your application.

Each hex digit encodes four bits.

Enter a (coeff of x), b, and c for the Linear Equation Calculator; use finite values in the indicated domain. The calculator reports x: Isolate x by subtracting b and dividing by a. A zero a (coeff of x) means the equation has either no unique solution or infinitely many.

Definition via change of base.

Change-of-base formula.

Enter a (1,1), b (1,2), c (2,1), and d (2,2) for the Matrix 2×2 Calculator; use finite values in the indicated domain. The calculator reports Determinant, Inv a, Inv b, Inv c, and Inv d: det = ad−bc; inverse is (1/det) times the adjugate. An inverse exists only when Determinant is nonzero.

Enter base a, exponent e, and modulus m (non-zero). Related tools: Extended GCD / Modular Inverse and Toy RSA Demonstrator.

Divide the part by the whole, then multiply by 100.

What is P% of X? Multiply the fraction of 100 by X.

X is what percent of Y? Divide the part by the whole, then scale to 100.

Percent change from A to B measures how much the value grew or shrank relative to the starting amount.

Divide the absolute difference by the absolute theoretical value, then scale by 100.

Permutations.

Combinations.

Enter Integer ≥ 2 for the Prime Factors Calculator; use finite values in the indicated domain. The calculator reports Prime factors and Factor count: Prime factorization writes a number as a product of primes. The input must be a whole number of at least 2.

Enter a, b, and c for the Proportion Calculator; use finite values in the indicated domain. The calculator reports x: Cross-multiply to solve a proportion. a must be nonzero because it is the divisor used to recover x.

Enter a (x²), b (x), and c for the Quadratic Equation Calculator; use finite values in the indicated domain. The calculator reports Discriminant, Root 1, and Root 2: Roots come from the quadratic formula using the discriminant. Set a (x²) nonzero; a negative Discriminant has no real Root 1 or Root 2.

Remainder after division.

Scale, round to nearest integer, then scale back.

Expressions use standard precedence with right-associative powers. Trig and inverse trig respect the DEG/RAD mode; hyperbolic functions are always in radians.

Scale so the first digit is in a known place, round, then scale back.

Two-point slope

x₂ ≠ x₁. Same units on each axis.

Horizontal line

y₂ = y₁ and the points are distinct.

Vertical line

Slope is undefined; do not divide by zero.

Point-slope form

A line through (x₁, y₁) with slope m.

Slope-intercept

b is the y-intercept when the line crosses x = 0.

Pick distinct small primes p and q, a public exponent e coprime to φ(n), and a message integer m with 0 ≤ m < n. Related: Modular Exponentiation, Extended GCD / Modular Inverse, Euler Totient.

Never use these keys for real security; production RSA uses large random primes and audited libraries.

For bit width w, values range from −2^(w−1) to 2^(w−1)−1; negative numbers wrap modulo 2^w.

Multiply the whole by P/100.