Modular Arithmetic

MODULAR ARITHMETIC: 

Modular Arithmetic is the mathematics of remainders. 

 if m l a-b the number m is called the modulus. 

A common example for Modular Arithmetic is Clock Arithmetic: 

There is an additive law to modular arithmetic that says: 

There is also a commutative law to modular arithmetic that says: 

Example: What is 38 mod 12?

So what you do is divide 12 into 38 and see what the remainder is, 

Example: What is 59 mod 4? 

Examine Question: What is 83 mod 5? 

There are two different tricks to finding mod's the first method is called the casting out nines method: 

the casting out nines method states that for a positive integer x, we define S(x) to be the sum of the digits of x. 

Example: What is S(5409)?

So we need to add all the number in S(x) together and we get 5+4+0+9=18

Now we claim: that 

so using our claim we get 

Examine Question: What is S(1688)?

The second method is just like the casting out of the nines method but using mod 11. So for a positive integer x, we define S(x) to sum of the digits of x. 

Example: What is S(13748)? 

1-3+7-4+8=13

Now we claim that

so using our claim we get 

Examine Question: What is S(34693)?

Work cited to: our lecture notes