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This is the Solved Exam of Number Theory which includes Last Complete Solution, Prime, Integer Relatively, Root Modulo, Distinct Primes, Primitive Root Modulo, Discrete Logarithms, Incongruent etc. Key important points are: Integers , Explicit Factorization, Information, Non Zero Polynomial, Integer CoeCients, Integer, Depending, Complex CoeCients, Polynomials, Complex Numbers
Typology: Exams
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The point value of each problem is given in the margin.
(10) 1. Find integers q, r such that −25 = 6q + r with a) 0 ≤ r < 6.
b) |r| ≤ 3.
(10) 2. Use the Euclidean Algorithm to find the greatest common divisor of 42 and
(14) 3. Find the general solution of the linear equation 46x − 28 y = 6.
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(12) 4. Use properties of congruences to compute the least residue of the following numbers modulo 7. (Avoid long multiplication in Z.) (a) 707 · 145 − 17
(b) 78^2 + 72^5
(12) 5. Prove the following theorem. If a, b, c are integers such that a|bc and (a, b) = 1 then a|c.
(18) 6. True, False. Circle T or F. True means that the statement is true for all choices of integers a, b, c, d. (a, b) =GCD. [a, b] =LCM.
T F a) For any integer a, 0|a. T F b) If a|b and a|c then a|(2b − c). T F c) If a|bc then either a|b or a|c. T F d) If d|a and d|b then d|[a, b]. T F e) For any integer q, (17 − 5 q, 5) = 1 T F f) If 6|(a + b) then a ≡ −b (mod 6). T F g) For any a, the equation 3x − 6 y = a has a solution in Z. T F h) Any integer can be expressed as a linear combination of 7 and 11. T F i) If f 1 = 1, f 2 = 1, f 3 = 2, ... is the Fibonacci sequence, then for any positive integer n, fn + f 2 n = f 3 n.
