gauss student sample problems: solutions
... For the Pan Australian Games the three states at the top of the medals table (Victoria, S.A. and W.A.) had won a total of 186 medals. Victoria had won the most gold medals and S.A. had as many gold medals as bronze medals. Victoria and S.A. won the same number of silver medals. W.A. had two more sil ...
... For the Pan Australian Games the three states at the top of the medals table (Victoria, S.A. and W.A.) had won a total of 186 medals. Victoria had won the most gold medals and S.A. had as many gold medals as bronze medals. Victoria and S.A. won the same number of silver medals. W.A. had two more sil ...
Algorithmic Number Theory
... Note: It is not necessary for q and r chosen in the above theorem to be the quotient and remainder obtained by dividing b into a. The theorem holds for any integers q and r satisfying the equality a = bq + r. The Euclidean theorem directly gives us an efficient algorithm to compute the GCD of two nu ...
... Note: It is not necessary for q and r chosen in the above theorem to be the quotient and remainder obtained by dividing b into a. The theorem holds for any integers q and r satisfying the equality a = bq + r. The Euclidean theorem directly gives us an efficient algorithm to compute the GCD of two nu ...
Chapter 4 The Group Zoo
... identity element 1̄: a · 1̄ = a. But not every a has an inverse! For an inverse ā−1 of ā to exist, we need aa−1 = 1 + zn, where z ∈ Z. Example 7. If n = 4, 2 cannot have an inverse, because 2 multiplied by any integer is even, and thus cannot be equal to 1 + 4z which is odd. To understand when an ...
... identity element 1̄: a · 1̄ = a. But not every a has an inverse! For an inverse ā−1 of ā to exist, we need aa−1 = 1 + zn, where z ∈ Z. Example 7. If n = 4, 2 cannot have an inverse, because 2 multiplied by any integer is even, and thus cannot be equal to 1 + 4z which is odd. To understand when an ...
Practice Midterm Solutions
... 5. Suppose a is an integer coprime to 7. The claim is that for any integer b, gcd(a, b) = gcd(a, 7b). As in problem 1, we proceed by showing that the common divisors of a and b are precisely the common divisors of a and 7b. First, we must show that if d divides a and d divides b, then d divides a a ...
... 5. Suppose a is an integer coprime to 7. The claim is that for any integer b, gcd(a, b) = gcd(a, 7b). As in problem 1, we proceed by showing that the common divisors of a and b are precisely the common divisors of a and 7b. First, we must show that if d divides a and d divides b, then d divides a a ...
Student_Solution_Chap_09
... a. Mersenne defined the formula Mp = 2p − 1 that was supposed to enumerate all primes. However, not all Mersenne numbers are primes. b. Fermat defined the formula Fn = 22n + 1 that was supposed to enumerate all primes. However, not all Fermat’s numbers are primes. 9. We mentioned the trial-division, ...
... a. Mersenne defined the formula Mp = 2p − 1 that was supposed to enumerate all primes. However, not all Mersenne numbers are primes. b. Fermat defined the formula Fn = 22n + 1 that was supposed to enumerate all primes. However, not all Fermat’s numbers are primes. 9. We mentioned the trial-division, ...
