MMW | Module 5.1 & 5.3 THE DIVISION ALGORITHM & RESIDUE CLASSES, and PROPERTIES OF CONGRUENCE

a = bq + r

In general, if we divide an integer a by an integer b, we get?

Positive

a = bq + r

where b is _______ and the existence of q and r are such that r is strictly less than b but greater than or equal to 0.

Greater than or equal to 0

a = bq + r

where b is positive and the existence of q and r are such that r is strictly less than b but ?

Strictly less than

where b is positive and the existence of q and r are such that r is __________ b but greater than or equal to 0.

Dividend

a = bq + r

a is?

Divisor

a = bq + r

b is?

Quotient

a = bq + r

q is?

Remainder

a = bq + r

r is?

a = bq + r

This is the Division Algorithm. It describes the process in the long division but it will not tell us how to find the quotient and the remainder.

Division Algorithm

a = bq + r

This is the __________. It describes the process in the long division but it will not tell us how to find the quotient and the remainder.

Residue Class

When dividing by b, the division algorithm informs us that there could be b different remainders. We can group integers by the remainder if we fix this divisor. Each group is referred to as a?

Remainder Class Modulo b

Residue Class is also known as?

Congruence

Comparing remainders of two integers is another interesting topic in the field of mathematics particularly in Number Theory. The term for comparing remainders of two integers is called?

Equivalence Relation

CONGRUENCE mod m is an?

Congruence

Let m be a positive integer. Two integers a and b are congruent modulo m if they each have the same remainder on division by m. If this is so, then we write:

a ≡ b (mod m)

Such a statement is called a?

Equivalence Relation

CONGRUENCE mod m is an?

Reflexive Property

PROPERTY: a ≡ a (mod m)

Symmetric Property

PROPERTY: If a ≡ b (mod m), then b ≡ a (mod m).

Transitive Property

PROPERTY: If a ≡ b (mod m)

and b ≡ c (mod m), then a ≡ c (mod m).