Lesson 4, Topic 1
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# The Cyclic Event

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A good example of cyclic event is the 12-hour clock. In the 12-hour clock arithmetic, the hour hand can move in both directions as shown the diagram

Example 1

If the time is presently 9 oâ€™clock, what would the time be in   (i) 92 hours (ii) 750 hoursâ€™ time?

Solution

(i) 92 hours = 92 Ã· 12 = 7 r 8

= ( 12 x 7) hrs + 8 hrs

= 9 oâ€™clock + 8hrs

= 5 oâ€™clock

(ii)750 hours = 750 Ã· 12 = 62 r 6

= (12 x 62)hrs + 6 hrs

= 9 oâ€™clock + 6 hrs

= 3 oâ€™clock

Example 2

If the time is presently 3 oâ€™clock, what time was it (i) 9 hours ago (ii) 25 hours ago?

Solution:

(i) Counting in the anti-clockwise direction, i.e. we have to count 9 spaces backward i.e. the time was 6 oâ€™clock.

(ii) 25 = (12 x 2)hrs + 1 hr

Hence, counting one step backward (or in an anticlockwise direction), we have (3 â€“ 1) = 2

i.e.  2 oâ€™clock

The clock arithmetic is synonymous with Modular Arithmetic which is by way of wrapping numbers around after they reach a certain value.

Modular Arithmetic system involves integers only. When an integer is divided by 5, the possible remainders are 0, 1, 2, 3, and 4; we thus refer to these numbers as the residues of modulo 5 or mod5.

Similarly, 0, 1, 2, 3, 4, 5 consist of the set of residues for mod 6.

In general, {0, 1, 2, 3, 4, 5, 6, …… n – 1} is called the set of residues for mod n.

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