9 methods · 22 questions
To control an equation or a divisibility over the integers, project it modulo a carefully chosen so that the unknowns can only take a few residues. Squares, cubes and powers occupy very few residue classes, so testing all cases of often forces a variable to be divisible by , fixes a parity, or kills the equation outright.
Trigger: an equation/divisibility in integers where a power, a parity, or a "last digit" is involved, and you want to forbid or force solutions.
Instances:
(i) tabulate squares or cubes modulo to show only certain residues occur.
(ii) read "ends in digit " as a statement modulo or modulo a power of .
(iii) reduce a congruence cipher modulo the stated base to recover unknowns.
(iv) split on parity (the case ) before the general modulus.
Linked questions (3)