Deck 4: Elementary Number Theory and Methods of Proof

Question 1

Prove the following statement: For all real numbers x,

Answer 1

Question 2

Prove the following statement: There is no smallest positive rational number.

Answer 2

Question 3

Prove the following statement by contradiction: For all real numbers r and s, if r is rational
and s is irrational, then r + 2s is irrational.

Answer 3

Question 4

Outline a proof of the following statement by writing the "starting point" and the "conclusion
to be shown" in a proof of the statement.
That is, complete the sentences below.

Question 5

Is 605.83 a rational number? Justify your answer.

Question 6

Is 56.745 a rational number? Justify your answer.

Question 7

Consider the following statement:
(a) Write a negation for Statement A.
(b) Disprove Statement A. That is, show that Statement A is false.

Question 8

Find a counterexample to show that the following statement is false:

Question 9

Prove the following statement: The sum of any two consecutive integers can be written in the
form 4n + 1 for some integer n.

Question 10

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.

Question 11

Show that the following statement is false: The product of any two irrational numbers is
irrational.

Question 12

Is 0 divisible by 3? Justify your answer.

Question 13

State precisely (but concisely) what it means for an integer n to be odd.

Question 14

Question 15

State precisely (but concisely) what it means for a number r to be rational.

Question 16

State precisely (but concisely) what it means for an integer n to be divisible by an integerd.

Question 17

Question 18

Prove the following statement 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.

Question 19

Does 12 divide 72? Justify your answer.

Question 20

Prove the following statement 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.

Question 21

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.

Question 22

True or false? For any irrational number r, r2 is irrational. Justify your answer.

Question 23

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.

Question 24

Prove by contradiction that
is irrational.)

Question 25

Use the Euclidean algorithm to find the greatest common divisor of 284 and 168. Show your
work.

Question 26

Question 27

Consider the following statement: For all integers n, if n3 is even then n is even.
(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.

Question 28

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