Practice question · Select all that apply
Select every operation you may perform on congruences modulo n with no extra conditions.
Hints
- Four of these follow from congruence being compatible with arithmetic; one is the famous exception.
- Try cancelling the 2 in '6 is congruent to 2 mod 4' and see whether the result survives.
Show the answer
- A. Replace any number by its remainder mod n at any stage
- B. Subtract two congruences
- C. Multiply two congruences
- E. Add two congruences
Why
Addition, subtraction, multiplication and reduce-as-you-go are all safe. Cancellation is not: 6 is congruent to 2 mod 4, but cancelling 2 gives '3 congruent to 1 mod 4', which is false. Division mod n needs a multiplicative inverse, and those exist only for numbers coprime to n.
Practise Modular Arithmetic
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More questions on Modular Arithmetic
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