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The remainders in the period, which are 3, 2, 6, 4, 5, 1, form a rearrangement of all nonzero remainders modulo 7, implying that 3 is indeed a primitive root modulo 7. This derives from the fact that a sequence ( g k modulo n ) always repeats after some value of k , since modulo n produces a finite number of values.
Even without knowledge that we are working in the multiplicative group of integers modulo n, we can show that a actually has an order by noting that the powers of a can only take a finite number of different values modulo n, so according to the pigeonhole principle there must be two powers, say s and t and without loss of generality s > t, such that a s ≡ a t (mod n).
When the "example for laypeople" was first added, the first four lines of the example were fairly obvious, but the reasons for choosing the congruent values "3 ×3 4 (mod 7)≡3 ×4" on the fifth line and "(3 3) 2 ≡ 6 2 (mod 7)" on the sixth line were completely unobvious.
Kraft's résumé is a bit different. The 83-year-old purchased a controlling stake in the Patriots in 1994 and has since presided over the most successful franchise of the 21st century.
There’s also CNN’s New Year’s Eve Live with Anderson Cooper and Andy Cohen, beginning at 8 p.m. ET on CNN (and streaming live on Max) and New Year's Eve Live: Nashville's Big Bash! hosted by ...
(Reuters) -U.S. President-elect Donald Trump on Wednesday asked a federal judge to dismiss a lawsuit accusing him of making defamatory statements during his campaign about five Black and Hispanic ...
The first series of Intellecs included the Intellec 4 for the 4004, the Intellec 4 Mod 40 for the 4040, the Intellec 8 for the 8008, and the Intellec 8 Mod 80 for the 8080. [2] The Intellec 4 and 8 were introduced at the June 1973 National Computer Conference in the New York Coliseum. [3]