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#number_theory
#congruences
#modulo
#mod
#congruent
#a=b(modm)
Another approach to divisibility questions is through the arithmetic of remainders,
or the theory of congruences as it is now commonly known. The concept, and
the notation that makes it such a powerful tool
Definition :. Let n be a fixed positive integer. Two integers a and b are said to be
congruent modulo n, symbolized by
a ≡ b (mod n)
if n divides the difference a − b; that is, provided that a − b = kn for some
integer k
To fix the idea, consider n = 7. It is routine to check that
3 ≡ 24 (mod 7) − 31 ≡ 11 (mod 7) − 15 ≡ −64 (mod 7)
because 3 − 24 = (−3)7, −31 − 11 = (−6)7, and −15 − (−64) = 7 · 7. When
n | (a − b), we say that a is incongruent to b modulo n, and in this case we write
#التطابقات
#باقي_القسمة
#aum