Problem (2), 10 points Disprove the following statement: If m > 2 is an integer, then...
(3 points) How many integer solutions of + +33 +34 = 30 are there with I, > 0, 2 <5, 12 <11, 13 < 13, 14 < 23?
Problem 3 Let n and k > l be positive integers. How many different integer solutions are there to x1 +...+ In = k, with all xi <l?
4. (10 points) Prove that the following statement is false. There exists an integer k > 4 such that k is a perfect square and k – 1 is prime.
Prove or Disprove that:
If a > 0 and b are two rational numbers, then a' is a rational number.
Prove that is an integer for all n > 0.
How to prove/disprove
is surjective?
F:2*Z >Zf (m, п) т
please answer the question by computer
Problem 2. (25 points) Consider the following integer nonlinear programming problem. max 2 = x1x2x s.t. X1 + 2x2 + 3x3 < 10, X1 >1, x2 > 1, X3 >1, X1, X2, X3 are integers. Use dynamic programming to solve this problem.
Problem 7: Prove that for all integers n > 2, n+1 n 10-11 - n n +
prove by induction!
Ex 5. (15 points total] For a natural integer n > 2, define n := V1+V1+ V1 +.. n times For instance ra = V1 + V1+V1+vī. (5a) (5 points) Write ræ+1 in function of In. (5b) (10 points) Prove that for all natural integers n > 2, In & Q.
5. (a) Show that 26 = 1 mod 9. (b) Let m be a positive integer, and let m = 6q+r where q and r are integers with 0 <r < 6. Use (a) and rules of exponents to show that 2" = 2 mod 9 (c) Use (b) to find an s in {0,1,...,8} with 21024 = s mod 9.