Problem 3 Let n and k > l be positive integers. How many different integer solutions...
1. Fix n and k. How many positive integer solutions are there to x1 + + xk = n where xi i for all i?
Let n be a positive integer. For each possible pair i, j of integers with 1 sisi<n, find an n x n matrix A with the property that 1 is an eigenvalue of A with g(1) = i and a(1) = j.
(3 points) How many integer solutions of + +33 +34 = 30 are there with I, > 0, 2 <5, 12 <11, 13 < 13, 14 < 23?
Exercise 7. Let X be a standard normal random variable. Prove that for any integer n > 0, ELY?"] = 1207) and E[x2n+] = 0.
Problem (2), 10 points Disprove the following statement: If m > 2 is an integer, then (a + b) (mod m) = a (mod m) +b (mod m) for all integers a and b.
Let n be a positive integer. For each possible pair i, j of integers with 1<i<i <n, find an n xn matrix A with the property that 1 is an eigenvalue of A with g(1) = i and a(1) = j.
question 5 5. (a) Informally find a positive integer k for which the following is true: 3n + 1 < n2 for all integers n > k-4 (b) Use induction to prove that 3n +1 < n2 for all integers n 2 k. 6. Consider the following interval sets in R: B-4.7, E = (1,5), G = (5,9), M-[3,6]. (a) Find (E × B) U (M × G) and sketch this set in the-y plane. (b) Find (EUM) x (BUG)...
Problem 7: Prove that for all integers n > 2, n+1 n 10-11 - n n +
1. Let n be a positive integer with n > 1000. Prove that n is divisible by 8 if and only if the integer formed by the last three digits of n is divisible by 8.
Say that a < 0) and k is a positive integer. Find a constant c such that tk edt < ceżat for all t > 0.