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

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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
6x3 (mod 9) 6x8 (mod 10) 6x2 (mod 11)
smallest
next
pf3
pf4

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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