Multiplicative Inverses in Congruences
Recall the Cancellation property of congruences:
m ).
ac ≡ bc (mod m) ⇒ a ≡ b (mod (c,m
)
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This means that in general, division is not an
allowable operation for congruences: if
ac ≡ bc €
(mod m), we may not be able to conclude that
a ≡ b (modm). This is possible only if (c, m) = 1.
In other words, numbers have multiplicative
inverses mod m only when they are relatively
prime to m. How do we tell? By using the
Euclidean algorithm, we can check that (c, m) = 1,
but by using the extended Euclidean algorithm, we
can actually find the inverse of c:
(c, m) = 1 ⇒ ∃x, y (cx + my = 1)
⇒ ∃x, y (cx + my ≡ 1 (mod m))
⇒ ∃x, y (cx ≡ 1 (mod m))
whence x is the multiplicative inverse of c mod m.
(Recall that while the integer x is not a unique
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solution
to the Diophantine equation cx + my = 1,
all other solutions are conrguent to x mod m, so the
congruence class of x is unique; that is, c has a
unique multiplicative inverse as a congruence class
mod m.)
We can now extend our knowledge about solving
linear Diophantine equations to the solution of
linear congruences.
Fundamental Theorem of Linear Congruences
The linear congruence ax ≡ b (mod m) is solvable
only if (a, m)|b. When it does have solutions, the
congruence has exactly (a, m) congruence classes of
solutions: if x0 is one congruence class that solves
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the
congruence,
all others have the form
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€m
x = x0 + (a,m)
k, for k = 0,1,K, (a,m) − 1.
Proof Solving the congruence ax ≡ b (mod m) is
equivalent to solving the linear Diophantine
€equation ax + my = b. So by Brahmagupta’s
Theorem, the congruence is solvable precisely when
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(a, m)|b, and has solutions of the desired form
m k for integral values of k. These
x = x0 + (a,m)
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solutions are distinct mod m for exactly the (a, m)
values indicated above. //
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