•Let a and b
be
integers and m be a positive integer. We say that a
is congruent to b modulo m
if
m divides a – b.
m divides a – b.
•We
use the notation a ≡ b (mod m) to
indicate that a is congruent to b modulo m.
•In
other words:
a ≡ b (mod m) if and only if a mod m = b mod m.
a ≡ b (mod m) if and only if a mod m = b mod m.
•Is it
true that 46 º 68
(mod 11) ?
•Yes,
because 11 | (46 – 68).
•Is it
true that 46 º 68
(mod 22)?
•Yes,
because 22 | (46 – 68).
•For which
integers z is it true that z º 12
(mod 10)?
•It is
true for any zÎ{…,-28,
-18, -8, 2, 12, 22, 32, …}
Theorem:
Let m be a positive integer. The integers a
and b are congruent modulo m if and only if there is an integer k such that a =
b + km.
7 ≡
1(mod3)
−8 ≡
1(mod3)
−2 ≡
1(mod3)
7 ≡
−8(mod3)
−2 ≡
7(mod3)
Theorem:
Let m be
a positive integer. If
a º b (mod m) and c º d (mod m), then
a
+
c ≡ b + d (mod m) and
ac ≡ bd (mod m).
Proof:
We know that a ≡ b (mod m) and c ≡ d (mod m) implies that there are
integers s and t with
b = a + sm and d = c + tm.
b = a + sm and d = c + tm.
Therefore,
b + d = (a + sm) +
(c + tm) = (a + c) + m(s + t) and
bd = (a + sm)(c + tm) = ac + m(at + cs + stm).
Hence, a + c ≡ b + d (mod m) and ac ≡ bd (mod m).
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