{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# Mod - Modular Arithmetic Denition If a b Z and m Z we say...

This preview shows pages 1–2. Sign up to view the full content.

Modular Arithmetic Definition: If a, b Z , and m Z + , we say that a is congruent to b modulo m , and write a b (mod m ) if a and b leave the same remainder on division by m . Examples . 10 24 (mod 7), - 15 12 (mod 9), and 442 2 (mod 10). You can think of the integers (mod m ) as the hours on a circular clock with m hours, 0 , 1 , . . . , m - 1. For positive numbers you move around the circle clockwise, and for negative numbers you move counter-clockwise. The important part is that the number of times you go around the circle and return to zero makes no difference to where you end up. What matters is the number of places you move when it is no longer possible to make it around the circle any more, and this number is one of 0 , 1 , 2 , . . . , m - 1. These are the possible remainders on division by m . The universe of integers (mod m ) really only consists of the numbers 0 , 1 , 2 , . . . , m - 1; modulo m , any other integer is just one of these with another name. In many programming languages there is a function mod . If m = 0 is an integer, then a (mod m ) is the unique number among 0 , 1 , 2 , . . . , m - 1 to which a is congruent modulo m . It is the remainder (as in the division algorithm - that’s why its unique) when a is divided by m . Questions . Prove that a non-negative integer is congruent to its last digit (its ones digit) (mod 10). What is a non-negative integer congruent to (mod 100)? (mod 1000)?

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

Mod - Modular Arithmetic Denition If a b Z and m Z we say...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online