Exam 4: Elementary Number Theory and Methods of Proof

Is 0 divisible by 3? Justify your answer.

Consider the following statement: For all real numbers r, if r3 is irrational then r is irrational. (a) Prove the statement either by contradiction or by contraposition. Clearly indicate which method you are using. (b) If you used proof by contradiction in part (a), write what you would "suppose" and what you would "show" to prove the statement by contraposition. If you used proof by contraposition in part (a), write what you would "suppose" and what you would "show" to prove the statement by contradiction.

Is 605.83 a rational number? Justify your answer.

Consider the following statement: For all integers n, if n3 is odd then n is odd. (a) Prove the statement either by contradiction or by contraposition. Clearly indicate which method you are using. (b) If you used proof by contradiction in part (a), write what you would "suppose" and what you would "show" to prove the statement by contraposition. If you used proof by contraposition in part (a), write what you would "suppose" and what you would "show" to prove the statement by contradiction.

Find a counterexample to show that the following statement is false: $\text { For all nonzero real numbers } a , b , c \text { and } d , \frac { a } { b } + \frac { c } { d } = \frac { a + c } { b + d } \text {. }$

Prove the statement below directly from the definitions of the terms. Do not use any other facts previously proved in class or in the text or in the exercises. $\text { For all integers } n , n ^ { 2 } + n + 1 \text { is odd. }$

Use the Euclidean algorithm to calculate the greatest common divisor of 10,673 and 11,284. Show your work.

Prove the following statement: For all real numbers x, $\lfloor x - 2 \rfloor = \lfloor x \rfloor - 2$