In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain valuethe 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 ).

Based on the article and the following list of answers, write a list of questions.

1. 12
2. divided into two 12-hour periods
3. Carl Friedrich Gauss
4. 1801.
5. Disquisitiones Arithmeticae",
6. Yes
7. 0
8. Three
9. clock time
10. In mathematics, modular arithmetic is a system of arithmetic for integers,
11. In mathematics
12. No
13. numbers
14. 0:00
15. unknown
16. no
17. no
18. Modulo operation
19. no
20. Carl Friedrich Gauss
1. How many hours are on the clock?
2. how is the day divided?
3. who published it?
4. When?
5. in what book?
6. Does 12 equal anything else?
7. What?
8. What time is it 8 hours after 7?
9. what wraps around after 12 hours?
10. what is modular arithmetic?
11. in what study?
12. do letters wrap around?
13. then what does
14. What can 12:00 also mean?
15. Does 3:00 equal 18:00?
16. are the intergers incompatiable?
17. Is the integer negative?
18. what does mod mean?
19. Can it be any number?
20. who developed the modern tech?