Modular Arithmetic

Explain how negative integers can be combined under the operations of addition and multiplication using mod 5

For combining negative integers under the addition operation the following formula stated below should be used Suppose -6 -1mod5 and -7 -2 mod5

Therefore -6 (-7 -1 (-2 ) mod5

-13 -3mod5

When checking the work , dividing -13 by 5 gives a remainder of -3 or -13- (-3 -10 which is divisible by 5 (William Stein ,2007

For combining negative integers under the multiplication operation the following formula stated below should be used p

Suppose -6 -1mod5 and -7 -2 mod5

Therefore -6 -7 -1 -2 mod5 2 mod5

This can be checked by multiplying -6 and -7 which gives 42 , and dividing this number by 5 gives a remainder of 2 Examples for multiplication using positive integers

Example1

Suppose 14 4 mod5 and 18 3 mod5

2mod5

This can be checked by 14 18 252 and dividing this number by 5 leaves a remainder of 2

Example2

Suppose 21 1 mod5 and 7 2 mod5

2mod5

This can be checked by 21 7 147 and dividing this number by 5 leaves a remainder of 2

Examples for multiplication using negative integers

Example1

Suppose -6 -1 mod5 and -9 -4 mod5

4mod5

This can be checked by -6 -9 54 and dividing this number by 5 leaves a remainder of 4

Example2

Suppose -11 -1 mod5 and -8 -3 mod5

3mod5

This can be checked by -11 -8 88 and dividing this number by 5...