4. (10 points) Prove that the following statement is false. There exists an integer k >...
Fact: If d > 2 is an integer, then there exists a prime q such that q divides d. (1) Let e and f be positive integers. Prove that if ged(e, f) = 1, then god(e?,f) = 1. (2) Let m be a positive integer. Prove that if m is rational, then m is an integer.
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.
induction question, thanks. (15 points) Prove by induction that for an odd k > 1, the number 2n+2 divides k2" – 1 for all every positive integer n.
Prove that is an integer for all n > 0.
R->H 7. Prove by induction that the following equation is true for every positive integer n. (4 Points) 1. 4lk11tl + 2K ²+ 3k 4k+4+H26² +3k {(4+1) = (40k41) 40) j=1 (4i + 1) = 2 n 2 + 3n 2K?+75 +5 21 13 43 041) 262, ultz
4. An element a in a ring R is called nilpotent if there exists a non-negative integer n such that a" = OR (a) Let a and m > O be integers such that if any prime integer p divides m then pſa. Prove that a is nilpotent in Zm. (b) Let N be the collection of all nilpotent elements of a ring R. Prove that N is an ideal of R. (c) Prove that the only nilpotent element in...
2. (D5) Let n = o(a) and assume that a =bk. Prove that <a >=<b> if and only if n and k are relatively prime.
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.
Exercise 5. Prove that if f is a continuous and positive function on (0,1], there exists 8 >0 such that f(x) > 8 for any x € [0,1].
5. Suppose H and K are subgroups of G and H 10, and |K-21. Prove that 6. Consider the subgroup <3 > of Z12. Find all the cosets of < 3>. How many distinct cosets are there?