Congruence in the integers
Numbers with the same remainder on division by m are congruent. See how to calculate with them.
- France: grade 12
- Iran: grade 12
- Russia: grade 7
Guide
Every integer leaves 0, 1, 2 or 3 when divided by 4. Numbers with the same remainder form a class: 1, 5, 9, 13 and so on all leave 1.
If a natural number m divides a - b we say a and b are congruent modulo m and write a≡b. This means a and b leave the same remainder on division by m. For example 17≡2 mod 5 because 5|17 − 2.
You can add or subtract an integer on both sides of a congruence and multiply both sides by an integer. So a big number can be replaced by its remainder.
Uses: weekdays (mod 7), clock hours (mod 12) and check digits on numbers.
Work on the number chart or the answer box in each mission.