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?

Short answer

1. 6:00
2. 15:00
3. 19:00

Step-by-step solution

Concept of Division Algorithm 

• Let\(a\)be an integer and\(d\)be a positive integer.
• Then there are unique Integers\(q\)and\(T\)with\(0 \le \;r < d\)such that\(a = dq + r\).
• \(q\)is called the quotient and\(T\)is called the remainder.
• \(\begin{array}{*{20}{l}}{q = a{\rm{ div }}d}\\{\;r = a{\rm{ mod }}d}\end{array}\)

(a) Finding the time that 24-hour clock reads 100 hours after it reads 2:00

\(\begin{array}{l}a = 100\\d = 24\end{array}\)

Let us determine the quotient and remainder

\(\begin{array}{c}a = 100\\ = 96 + 4\\ = 4 \cdot 24 + 4\\ = 4\;d + 4\end{array}\)

The quotient is then the coefficient of\(d \Rightarrow q = 4\)

The remainder is the constant:\(r = 4\;({\rm{with}}\;0 \le 4 < 24)\)

\(100\;\bmod \;24 = 4\)

The remainder is 4, thus 100 hours after it reads 2:00 is the same time as 4 hours after it reads 2:00, 4 hours after 2:00 is 6 o’clock.

\(2:00 + 4:00 = 6:00\)

Hence the time is 6.00

 (b) Finding the time that 24-hour clock reads 45 hours after it reads 12:00

\(\begin{array}{l}a = 45\\d = 24\end{array}\)

Let us determine the quotient and remainder

\(\begin{array}{c}a = 45\\ = 24 + 21\\ = 1 \cdot 24 + 21\\ = 1\;d + 21\end{array}\)

The quotient is then the coefficient of\(d \Rightarrow q = 1\)

The remainder is the constant:\(r = 21\;({\rm{with}}\;0 \le 8 < 21)\)

\(45\;\bmod \;24 = 21\)

The remainder is 21, thus 45 hours before it reads 12:00 is the same time as 21 hours before it reads 12:00 or equivalently 3 hours after it reads 12:00, 3 hours after 12:00 or equivalently 3 hours after it reads 12:00. 3 hours after 12:00 is 15:00 (or 3 o’clock in the afternoon)

\(12:00 + 3:00 = 15:00\)

Hence the time is 15.00

(c) Finding the time that 24-hour clock reads 168 hours after it reads 19:00

\(\begin{array}{l}a = 168\\d = 24\end{array}\)

Let us determine the quotient and remainder

\(\begin{array}{c}a = 168\\ = 7.24\\ = 7 \cdot 24 + 0\\ = 7\;d + 0\end{array}\)

The quotient is then the coefficient of\(d \Rightarrow q = 7\)

The remainder is the constant:\(r = 0\;({\rm{with}}\;0 \le 0 < 24)\)

\(168\;\bmod \;24 = 0\)

The remainder is 0, thus 168 hours after it reads 19:00 is the same time as 0 hours after it reads 19:00. Thus the time will be 19:00 again

Hence the time is 19.00