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3.2. MODULAR ARITHMETIC 39
While % and ≡ are very closely related, they have an important difference:
% performs an operation and ≡ is a statement. The symbol % is an operation,
like addition. It turns two numbers into a third. But ≡ is a true/false statement,
like equals. The mathematical statement 7 = 9 is a false one, while 7 = 9 − 2 is
a true one. Similarly, 28 ≡ 2 (mod 26) is true and 29 ≡ 2 (mod 26) is false.
If A%n = B then A ≡ B (mod n). For example, 78%10 = 8 so 78 ≡ 8
(mod 10). Conversely, if A ≡ B (mod n), then A%n = B%n. For example,
31 ≡ 17 (mod 7) since 31%7 = 3 = 17%7. Notice, however, that 31%7 ≠ 17
and 17%7 ≠ 31. So the “equivalence modulo” statement, ≡ (mod n), is slightly
weaker than “ equal remainder,” %n =.
For the next several chapters the remainder operator will dominate and it
will be a while until we see how powerful equivalence is. But since we’ve done
most of the work, let us state the following.
Theorem 1 Suppose A and B are any integers, and n is a positive integer. We
write A ≡ B (mod n) if any of the following equivalent statements are true:
1. A%n = B%n.
2. A and B have the same remainder when divided by n.
3. n divides B − A with remainder 0.
4. A and B differ by a multiple of n.
Perhaps the language in the theorem is a bit unfamiliar. By “equivalent
statements” we simply mean that when numerical values are substituted for A,
B and n, then either all of the statements will be true, or all of them will be
false. In less formal language, each statement contains the same information,
they just present it in different ways. For example if A and B have the same
remainder when divided by n, then A − B will have no remainder. So A − B
will be a multiple of n, or A and B will differ by n.
Doing algebra using ≡ is very similar to the algebra you are used to. In fact,
+, −, × and the Associate, Commutative, and Distributive Rules all work using
≡ just like they always did with =. Division, however, is more complicated, as
the following examples show.
Examples: Examples of Division in Modular Arithmetic.
(1) 3x ≡ 9 (mod 7) has the usual solution x = 3.
(2) 3x ≡ 9 (mod 12) has the usual solution x = 3, but also x = 7 and x = 11.
(3) 3x ≡ 8 (mod 7) has the unusual solution x = 5.
(4) 3x ≡ 8 (mod 12) has no solutions.
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