algebra definitions

Question 1

set

Answer 1

A set is a collection of objects

Question 2

prime number

Answer 2

A natural number n∈ℕ is a prime number if n≠1 and the only positive factors of n are 1 and n

Question 3

factor

Answer 3

let a,b∈ℤ. we say that a is a factor of b if there exists z∈ℤ such that b = az

Question 4

composite number

Answer 4

A natural number n∈ℕ is called a composite number if n≠1 and there exists a,b∈ℕ with 1<a></a>

Question 5

common factor

Answer 5

Let a,b∈ℤ. A common factor of a is an integer c such that c|a and c|b

Question 6

highest common factor

Answer 6

Let a,b∈ℤ. The highest common factor of a and b is the largest integer h that is a common factor of a and b and is denoted h = hcf(a,b).

Unless a=b=0 and by convention we define hcf(0,0)=0

Question 7

comprime

Answer 7

Let a,b∈ℤ. We say a is comprime to b if hcf(a,b)=1

Question 8

common multiple

Answer 8

A common multiple of a and b is an integer m such that a|m and b|m

Question 9

lowest common multiple

Answer 9

The least common multiple of a and b is the smallest l∈ℕ that is a common multiple of a and b and is denoted l=lcm(a,b); unless one of a or b is equal to 0 and then by convention we define lcm(a,0)= lcm(0,b)=0

Question 10

perfect nth power

Answer 10

we say that a∈ℕ is a perfect nth power is there exists b∈ℕ s.t. a=bⁿ

Question 11

congruent

Answer 11

Let n∈ℕ and a,b∈ℤ. We write a≡bmodn and say that a is congruent to bmodn if n|a-b

Question 12

linear congruence equation

Answer 12

A linear congruence equation is an equation of the form ax≡bmodn where n∈ℕ and a,b∈ℤ and we are trying to solve for x

Question 13

congruence class

Answer 13

let n∈ℕ and a∈ℤ. We define the congruence class of a modulo n to be [a]ₙ = {x∈ℤ: x≡amodn}