3.7 Clock Arithmetic

To determine the value of a number modulo 12, divide the number by 12 and record the remainder.

1. To find the value 34 modulo 12:
   

   Step 1: Determine the remainder when 34 is divided by 12 using long division. The largest multiple of 12 that is less than or equal to 34 is 24, which is the product of 12 and 2.

   $$\begin{array}{c}12\stackrel{2}{\overline{)34}}\\ \underset{\_}{24}\end{array}$$

   Step 2: Performing the subtraction yields 10.

   $$\begin{array}{c}12\stackrel{2}{\overline{)34}}\\ \underset{\_}{24}\\ 10\end{array}$$

   Since that subtraction resulted in a number less than 12, that is the remainder, 10. The value of 34 modulo 12 is 10, or 34 = 10 (mod 12).

2. To find the value 539 modulo 12:
   

   Step 1: Determine the remainder when 539 is divided by 12 using long division. We first look to the first two digits of 539, 53. The largest multiple of 12 that is less than or equal to 53 is 48, which is the product of 12 and 4.

   $$\begin{array}{c}12\stackrel{4}{\overline{)539}}\\ \underset{\_}{48}\end{array}$$

   Step 2: Performing the subtraction results in 5.

   $$\begin{array}{c}12\stackrel{4}{\overline{)539}}\\ \underset{5}{\underset{\_}{48}}\end{array}$$

   Step 3: Now, the 9 is brought down.

   $$\begin{array}{c}12\stackrel{4}{\overline{)539}}\\ \underset{59}{\underset{\_}{48}}\end{array}$$

   Step 4: The largest multiple of 12 that is less than or equal to 59 is once more 48 itself, which is $12\times 4$.

   $$\begin{array}{r}\begin{array}{c}12\stackrel{44}{\overline{)539}}\\ \underset{59}{\underset{\_}{48}}\end{array}\\ \underset{\_}{48}\end{array}$$

   Step 5: Finishing the process, the 48 is subtracted from the 59, yielding 11.

   $$\begin{array}{r}\begin{array}{c}12\stackrel{44}{\overline{)539}}\\ \underset{59}{\underset{\_}{48}}\end{array}\\ \underset{11}{\underset{\_}{48}}\end{array}$$

   We've used all the digits of 539, and the last subtraction resulted in a number less than 12, so that number, 11, is the remainder. The value of 539 modulo 12 is 11, or, 539 = 11 (mod 12).

3. To find the value 156 modulo 12:
   

   Step 1: Determine the remainder when 156 is divided by 12 using long division. We first look to the first two digits of 156, 15. The largest multiple of 12 that is less than or equal to 15 is 12 itself, which is the product of 12 and 1.

   $$\begin{array}{c}12\stackrel{1}{\overline{)156}}\\ \underset{\_}{12}\end{array}$$

   Step 2: Performing the subtraction results in 3.

   $$\begin{array}{c}12\stackrel{1}{\overline{)156}}\\ \underset{3}{\underset{\_}{12}}\end{array}$$

   Step 3: Now, the 6 is brought down.

   $$\begin{array}{c}12\stackrel{1}{\overline{)156}}\\ \underset{36}{\underset{\_}{12}}\end{array}$$

   Step 4: The largest multiple of 12 that is less than or equal to 36 is 36 itself, which is $12\times 3$.

   $$\begin{array}{r}\begin{array}{c}12\stackrel{13}{\overline{)156}}\\ \underset{36}{\underset{\_}{12}}\end{array}\\ \underset{\_}{36}\end{array}$$

   Step 5: Finishing the process, the 36 is subtracted from the 36, yielding 0.

   $$\begin{array}{r}\begin{array}{c}12\stackrel{13}{\overline{)156}}\\ \underset{36}{\underset{\_}{12}}\end{array}\\ \underset{0}{\underset{\_}{36}}\end{array}$$

We've used all the digits of 156, and the last subtraction resulted in a number less than 12, so that number, 0, is the remainder. The value of 156 modulo 12 is 0, or, 156 = 0 (mod 12).

We should note here that, had we been speaking of time, the 0 would be interpreted as 12:00.

To find that future time, we may determine the value of 89 (mod 12), either by long division or by using a calculator, such as Desmos. Then add the result to 3:00. Entering mod(89,12) in Desmos results in 5. Adding 5 hours, which was 89 (mod12), to 3:00 results in 8:00.

To find that past time, we may determine the value of 67 (mod 12), either by long division or by using a calculator, such as Desmos. Then subtract the result to 4:00. Entering mod(67,12) in Desmos results in 7. Subtracting 7 hours from 4:00 results in ‒3:00. We know, though, that time is not represented with negative times. This value, ‒3:00, indicates three hours before 12:00, which is 9:00. So, 67 hours before 4:00 was 9:00. We see this in the Figure 3.43.

Figure 3.43 Clock showing 7 hours subtracted from 4:00

We begin by multiplying 11 and 45, which is 495. Next, we find 495 modulo 12, either by dividing the result by 12 to determine the remainder, or by using a calculator. Entering mod(495,12) in Desmos yields 3. Had long division been used, the remainder would be 3. So $11\times 45=3$ modulo 12.

If you check your email every 5 hours nine times, that ninth check will occur 45 hours after 3:00, which is an addition of 45 hours to 3:00. So, we find 45 modulo 12, which is 9. Nine hours after 3:00 is 12:00. It will be 12:00 when you check your email the ninth time.

The first time you emptied the litter box was on a Thursday. So,the 10th time you empty the litter box will be 9 times later (you've already had your first turn, so 9 turns left!). This will happen 54 (9 times 6) days later. Finding the value of 54 modulo 7, using division to determine the remainder or using a calculator to find the value of 54 modulo 7 gives the answer 5. Five days after a Thursday is Tuesday.