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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.
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.
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 ...
The set {3,19} generates the group, which means that every element of (/) is of the form 3 a × 19 b (where a is 0, 1, 2, or 3, because the element 3 has order 4, and similarly b is 0 or 1, because the element 19 has order 2).
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.
9 mod 16 → 9, 25 mod 32 and 7 mod 16 → 7, 23 mod 16, while 1 mod 16 and 15 mod 16 don't lift to roots mod 32. For every k at least 3, there are four roots of x 2 − 17 mod 2 k , but if we look at their 2-adic expansions we can see that in pairs they are converging to just two 2-adic limits.