A Congruence Primer

prepared by:

Joseph Malkevitch
Mathematics Department
York College (CUNY)
Jamaica, New York 11451



web page:


Call two numbers congruent modulo m if they leave the same remainder when divided by a positive integer m other than 1, known as the modulus. The notation for numbers being congruent modulo m is due to Karl Friedrich Gauss, and resembles an equal sign, a symbol with which it shares many properties:



(Note that when one divides 17 or 2 by 5 the remainder is 2.)

(Note that when one divides 30 or 20 by 5 the remainder is 0.)

(Note that when one divides 5 into -2, the quotient is -1 and the remainder 3, that is: -2 = -1(5) + 3.)

In all three cases one can check that 5 divides the number on the right subtracted from the number on the left. Thus, for the third example, -2 - 3 = -5 which is divisible by 5.

This last observation is not an accident. An alternative point of view about congruence mod m is that (*) holds when a-b is divisible by m with a remainder of zero. It is easy to see that if a and b both leave the same remainder when divided by m, then a-b is divisible by m. Here is the proof. If a and b leave the same remainder when divided by m then we have a = qm + r and b = pm + r. Hence a - b = qm + r - (pm + r) = qm - pm = (q-p)m. Hence m divides a - b. The proof in the other direction is no more difficult.

Facts about congruences:

1. If


2. Every integer mod m is congruent to a number between 0 and (m-1). In particular every integer is congruent to a non-negative integer.




4. A similar result holds with multiplication replacing addition: if two pairs of numbers are congruent mod m then their products are congruent mod m.

Using the same numbers as in the example above we have: