Prove or disprove "Suppose that m and n are integers. If m > n ≥ 0, then gcd (m, n) = gcd (m − n, n)."
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Prove or disprove "Suppose that m and n are integers. If m > n ≥ 0, then gcd (m, n) = gcd (m − n, n)."
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- Find all solutions for the following pair of simultaneous congruences. 5x = 9 (mod 11) 3x = 8 (mod 13)7. For n 2 1, in how many out of the n! permutations T = (T(1), 7(2),..., 7 (n)) of the numbers {1, 2, ..., n} the value of 7(i) is either i – 1, or i, or i +1 for all 1 < i < n? Example: The permutation (21354) follows the rules while the permutation (21534) does not because 7(3) = 5. Hint: Find the answer for small n by checking all the permutations and then find the recursive formula depending on the possible values for 1(n).Determine φ (m), for m=12,15, 26, according to the definition: Check for each positive integer n smaller m whether gcd(n,m) = 1. (You do not have to apply Euclid’s algorithm.)
- Prove or disprove that for any x ∈ N, x(x+1)/2 ∈ N (where N = {0, 1, 2, 3, ….}Let N = {0, 1, 2, . . .} be the set of Natural Numbers. Given an n ∈ N, which of the followingconditions are necessary, and which of these conditions are sufficient, for the Natural Number,n, to be a factor of 10.(a) 1 is a factor of n.(b) 1 is a factor of 2n.(c) −n is a factor of 10.(d) 10 is a multiple of n.(e) n is divisible by 2.(f) n^2 is divisible of 5.(g) n = 10.1) Direct proofs: A. Prove that for any integer x, the integer x(x + 1) is even. B. Prove that n is even if and only if n^2 is even.
- Let a and b be integers. If 7| (17a + 256), prove that 7| (10a + 11b).2. Prove that 7L Σε B i=1 n² (n + 1)² 4Find the relation between the following functions: f(n) = log n and g(n) = Vn. (Square root for n)Hint: you may use L'Hopital's Theorem. For function f(n)=log n and time t=1 second, determine the largest size n of a problem that can be solved in time t, assume that the algorithm to solve the problem takes f(n) microseconds. Suppose you have algorithms with the two running times listed below. Suppose you have a computer that can perform 6 operations per second, and you need to compute a result in at most an hour of computation. For each of the algorithms, what is the largest input size n for which you would be able toget the result within an hour for:a) n^3b)10n^2