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For example, the expression "5 mod 2" evaluates to 1, because 5 divided by 2 has a quotient of 2 and a remainder of 1, while "9 mod 3" would evaluate to 0, because 9 divided by 3 has a quotient of 3 and a remainder of 0. Although typically performed with a and n both being integers, many computing systems now allow other types of numeric operands.
The extended Euclidean algorithm says that −299 ⋅ 10 + 3 ⋅ 997 = 1, so N′ will be 7. i ← 0 m ← 6 ⋅ 7 mod 10 = 2 j T c - ----- - 0 0485670 2 (After first iteration of first loop) 1 0485670 2 2 0485670 2 3 0487670 0 (After first iteration of second loop) 4 0487670 0 5 0487670 0 6 0487670 0 i ← 1 m ← 4 ⋅ 7 mod 10 = 8 j T c ...
In number theory, given a positive integer n and an integer a coprime to n, the multiplicative order of a modulo n is the smallest positive integer k such that (). [1]In other words, the multiplicative order of a modulo n is the order of a in the multiplicative group of the units in the ring of the integers modulo n.
For example, multiplication by 2 on Z/21Z has cycle decomposition (0)(1,2,4,8,16,11)(3,6,12)(5,10,20,19,17,13)(7,14)(9,18,15), so the sign of this permutation is (1)(−1)(1)(−1)(−1)(1) = −1 and the Jacobi symbol (2|21) is −1. (Note that multiplication by 2 on the units mod 21 is a product of two 6-cycles, so its sign is 1.
If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in 7 + 8 = 15, but 15:00 reads as 3:00 on the clock face because clocks "wrap around" every 12 hours and the hour number starts again at zero when it reaches 12. We say that 15 is congruent to 3 modulo 12, written 15 ≡ 3 (mod 12), so that 7 + 8 ≡ 3 ...
Modulo is a mathematical jargon that was introduced into mathematics in the book Disquisitiones Arithmeticae by Carl Friedrich Gauss in 1801. [3] Given the integers a, b and n, the expression "a ≡ b (mod n)", pronounced "a is congruent to b modulo n", means that a − b is an integer multiple of n, or equivalently, a and b both share the same remainder when divided by n.
Then () = means that the order of the group is 8 (i.e., there are 8 numbers less than 20 and coprime to it); () = means the order of each element divides 4, that is, the fourth power of any number coprime to 20 is congruent to 1 (mod 20).
Any character mod a prime power is also a character mod every larger power. For example, mod 16 [ 32 ] 1 3 5 7 9 11 13 15 χ 16 , 3 1 − i − i 1 − 1 i i − 1 χ 16 , 9 1 − 1 − 1 1 1 − 1 − 1 1 χ 16 , 15 1 − 1 1 − 1 1 − 1 1 − 1 {\displaystyle {\begin{array}{|||}&1&3&5&7&9&11&13&15\\\hline \chi _{16,3}&1&-i&-i&1&-1&i&i&-1 ...