Exam 2

Question 1

An integer n is ________ iff ∃k ε Z s.t. n = 2k + 1.

Answer 1

odd

Question 2

An integer n is ________ iff ∃k ε Z s.t. n = 2k.

Answer 2

even

Question 3

An integer n is ________ iff n > 1 and it has no positive divisor other than 1 and itself.
i.e. iff n > 1 and ∀r,s ε Z+ if n = r * s then r = 1 or s = 1.

Answer 3

prime

Question 4

An integer is __________ iff n > 1 and n is not prime.
i.e. iff n > 1 and ∃r,s ε Z+ s.t. n = r * s but r ≠ 1 and s ≠ 1.

Answer 4

composite

Question 5

r ε R is called a _____________ iff ∀a,b ε Z s.t. r = a/b and b ≠ 0.

Answer 5

rational number

Question 6

∀n ε Z ∀d ε Z s.t d ≠ 0 ______________, denoted d | n iff ∃k ε Z s.t. n = dk.

Answer 6

d divides n

Question 7

Any integer n s.t. n > 1 is either a prime or it can be uniquely written as a product of primes in a non-decreasing order.

Answer 7

The Fundamental Theorem of Arithmetic (FTA)

Question 8

∀n ε Z ∀d ε Z+ ∃! q, r ε Z s.t. n = dq + r and 0 ≤ r < d

Answer 8

Quotient Remainder Theorem