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8 Questions on the Congruence Theory - Examination 2 | MATH 302, Exams of Mathematics

Western Washington University (WWU)Mathematics

Material Type: Exam; Class: Introduction to Proofs Via Number Theory; Subject: Mathematics; University: Western Washington University; Term: Winter 2005;

Typology: Exams

Pre 2010

Uploaded on 08/17/2009

koofers-user-1dy
koofers-user-1dy 🇺🇸

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MATH 302 Test 2
February 15, 2005 Name
Put your answers in the space provided. Show your reasoning. Calculators are to be used when appro-
priate. The maximum score on the test is 30 points.
1. 4 points Use congruence theory to establish that 27|25n+1 + 5n+2 .
2. 4 points Find the two smallest positive solutions to
6x≡3 (mod 9) 6x≡8 (mod 10) 6x≡2 (mod 11)
smallest
next
pf3
pf4

Partial preview of the text

Download 8 Questions on the Congruence Theory - Examination 2 | MATH 302 and more Exams Mathematics in PDF only on Docsity!

MATH 302 Test 2February 15, 2005 Name

Put your answers in the space provided. Show your reasoning. Calculators are to be used when appro-priate. The maximum score on the test is 30 points.

  1. 4 points Use congruence theory to establish that 27 | 25 n+1^ + 5n+2.
  2. 4 points Find the two smallest positive solutions to 6 x ≡ 3 (mod 9) 6 x ≡ 8 (mod 10) 6 x ≡ 2 (mod 11) smallest next
  1. 4 points Prove: If a ≡ b (mod n) and c ≡ d (mod n), then a + c ≡ b + d (mod n).
  2. 4 points If a ≡ b (mod n), prove that gcd(a, n) = gcd(b, n).

7a. 1 point Exhibit primes larger than 20 of the form 6 k − 1 and 6 k + 1 Answers

7b. 1 point If p > 3 is a prime, then prove that p is either of the form 6 k − 1 or 6 k + 1.

7c. 2 points If p > 3 is a prime, then prove that p^2 + 2 is always composite.

  1. 3 points Find all incongruent solutions to 6 x ≡ 15 (mod 21) Answer