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).Let f (n) and g(n) be positive functions (for any n they give positive values) and f (n) = O(g(n)).Prove or disprove the following statement:
- 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.)6. Prove that if m and n are integers and mn is even, then m is even or n is even.5. Prove by simple induction onn that 2" > n
- 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.