In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus (plural moduli). The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book "Disquisitiones Arithmeticae", published in 1801. 

A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Usual addition would suggest that the later time should be , but this is not the answer because clock time "wraps around" every 12 hours. Because the hour number starts over after it reaches 12, this is arithmetic "modulo" 12. According to the definition below, 12 is congruent not only to 12 itself, but also to 0, so the time called "12:00" could also be called "0:00", since 12 is congruent to 0 modulo 12. 

Modular arithmetic can be handled mathematically by introducing a congruence relation on the integers that is compatible with the operations on integers: addition, subtraction, and multiplication. For a positive integer , two numbers and are said to be ", if their difference is an integer multiple of (that is, if there is an integer such that ). This congruence relation is typically considered when and are integers, and is denoted (some authors use instead of ; in this case, if the parentheses are omitted, this generally means that "mod" denotes the modulo operation, that is, that ).
How many hours are on the clock?
how is the day divided?
who published it?
When?
in what book?
Does 12 equal anything else?
What?
What time is it 8 hours after 7?
what wraps  around after 12 hours?
what is modular arithmetic?
in what study?
do letters wrap around?
then what does
What can 12:00 also mean?
Does 3:00 equal 18:00?
are the intergers incompatiable?
Is the integer negative?
what does mod mean?
Can it be any number?
who developed the modern tech?
12
o 12-hour periods
divided into two 12-hour periods
 divided into two 12-hour periods
Carl Friedrich Gauss
developed by Carl Friedrich Gauss
1801.
published in 1801.
Disquisitiones Arithmeticae",
his book "Disquisitiones Arithmeticae",
Yes
According to the definition below, 12 is congruent not only to 12 itself, but also to 0,
0
According to the definition below, 12 is congruent not only to 12 itself, but also to 0
Three
the time is 7:00 now, then 8 hours later it will be 3:00
clock time
but this is not the answer because clock time "wraps around"
In mathematics, modular arithmetic is a system of arithmetic for integers,
In mathematics, modular arithmetic is a system of arithmetic for integers,
In mathematics
In mathematics, modular arithmetic is a system of arithmetic for integers,
No
here numbers "wrap around" upon
numbers
where numbers "wrap around" upon reaching a certain value—the
0:00
could also be called "0:00", since 12 is congruent to 0 modulo 12. 
unknown
unknown
no
Modular arithmetic can be handled mathematically by introducing a congruence relation on the integers that is compatible with the operation
no
 For a positive integer , two numbers and are said to be 
Modulo operation
generally means that "mod" denotes the modulo operation, that is, that ).
no
where numbers "wrap around" upon reaching a certain value
Carl Friedrich Gauss
eveloped by Carl Friedrich Gauss 