Number theory step by step

Euclid — every division written out (1071 = 2·462 + 147, with the multiplication and subtraction beside it), the trace table (a, b, q, r), the gcd, the lcm and the reduced fraction. Below it, the same steps as tiling an a × b rectangle with the largest squares that fit. Lamé’s worst case (consecutive Fibonacci numbers) is pointed out.

Bézout — the divisions solved for their remainders, then back-substitution from the gcd line upwards (substitute, expand, collect), then the forward table (q, r, x, y with a check column) that code uses. When the gcd is 1 it reads off the inverse of b mod a. Fill in c to solve the congruence b·x ≡ c (mod a), including the case where the gcd is more than 1.

Jugs — fill, empty and pour between jugs of capacity A and B, or watch the shortest solution. The amounts strip shades every reachable amount (the multiples of gcd(A, B)), and when you hit the goal the tool counts your fills and empties as a Bézout combination.

Sieve — the sieve of Eratosthenes on a grid of any width, animated: each prime crosses out its multiples from p² in its own colour. Column headers show each column’s remainder; columns that share a factor with the width are marked, so width 6 shows why primes past 3 are 6k ± 1. The table of rounds matches the book’s worked example.

Trial division — d = 2, 3, 4, … with a table of every try, a number line with the divisor moving towards the √ marker (which jumps left as factors come out), and a plot of divisors tried against n, where the primes form the top envelope.

Clock — a dial for any modulus with x + y as a walk and x · y as y hops of x (extra laps drawn as inward rings), each with its division line. Chords joins every x to kx mod m: k = 2 on 200 points is a cardioid, 3 a nephroid. k may be fractional, and Sweep k animates it.

Mod tables — the multiplication (or addition) table mod m with inverses (the 1s) and zero divisors (0s in nonzero rows) coloured, the units listed with their inverses, and why you cannot cancel. Click a cell for its arithmetic.

% on negatives — truncated, floored and Euclidean division side by side with a = b·q + r checked, what each of 17 languages and functions prints, a chart of the remainder against a, and the two bugs it causes.

Fast powers — the exponent’s bits as switches, the book’s square-and-multiply table lit a bit at a time, the operation count against naive multiplication, and a race that runs the naive loop for real and estimates how long it would take. With no modulus it follows chapter 3’s version (no wasted last square).

Repeating decimals — long division with a remainder wheel: each step lights the remainder it lands on, the cycle turns red when a remainder comes round again, and the period is shown, with the conversion back to a fraction and a chart of the period of 1/q for q up to 50.

LCG — x → (a·x + c) mod m: the period and every cycle, the state graph (cycle in blue, tail in orange), the low bit’s pattern, and the scatter of consecutive pairs that shows the lattice. The three Hull–Dobell conditions are listed with Break it / Fix it buttons that change a or c as little as possible.

Check digits — Luhn, ISBN-10 and ISBN-13 as weighted tables with the sum and its remainder. Type ? for a digit to have it solved. For a valid code it tries every single-digit error and every swap of neighbours and lists the ones the scheme misses.

Diffie–Hellman — Alice, the public wire and Bob side by side, each power with its square-and-multiply table, the order of g, and Eve trying every exponent.

RSA — key generation with every step shown (prime checks, n and φ, gcd(e, φ) by Euclid, d by back-substitution and the forward table), encryption and decryption of a number or of a word letter by letter (repeated letters light up), and an attacker who factors n and times bigger keys.