Question

What time does a 24-hour clock read
(a) 100 hours after it reads 2:00?
(b) 45 hours before it reads 12:00?
(c) 168 hours after it reads 19:00?

Answer 1

We are given an example of arithmetic modulo 24. For example, it is 22:00 now, then in 5 hours the clock will show 3:00 instead of 27:00. In other words, we need to find the remainder of 27 when divided by 24 ($22+5 \equiv 3 \ (\mathrm{mod}\ 24)$).

1) Suppose it's 2:00. What time is it in 4 days and 4 hours ($4*24+4 =100$ hours)?$2+100 \equiv 2+4 \ (\mathrm{mod}\ 24)$. Answer: 6:00.

2) Suppose it's 12:00. What time was it 1 day and 21 hours ago ($1*24+21=45$ hours)?

$12-45 \equiv 12-21 \ (\mathrm{mod}\ 24)$

$-9 \equiv 15 \ (\mathrm{mod}\ 24)$. Answer: 15:00.

3) Suppose it's 19:00. What time is it in 7 days and 0 hours ($7*24=168$ hours)?

$19+168 \equiv 19 \ (\mathrm{mod}\ 24)$. Answer: 19:00.