What is $8\times 9$ in $12$ hour arithmetic?
Will it be $0$, or will it be $6$?
$8*9 = 72$ and $72/12 = 6$
So is the answer $0$ or $6$?
Asked
Active
Viewed 1,310 times
3
-
Is $12$ the same as $0$ or $1$ in clock arithmetic? – Erick Wong May 30 '13 at 05:47
-
3If instead of $72$ you had $73$ for which $73 / 12 = 6.08333333333\dots$, what would you say the answer is? – ShreevatsaR May 30 '13 at 06:01
2 Answers
5
Modular arithmetic deals with remainders upon division. The remainder of $72$ upon dividing by $12$ is $0$. So we have
$$8\cdot9\equiv72\equiv0\operatorname{mod}12$$
Jared
- 31,451
1
Remember, $$a\equiv b\bmod n\;\iff \;n\text{ goes into }a-b,$$ or in other words, $$a\equiv b\bmod n\;\iff \;\text{dividing }a-b\text{ by }n\text{ leaves no remainder}.$$ Using this definition, which statement is true: $$72\equiv 0\bmod 12$$ or $$\;\;\;72\equiv 6\bmod 12 \;?$$
Zev Chonoles
- 129,973