7 DAY CLOCK ARITHMETIC

In modular arithmetic, if the divisor is 7, it is called as  hour clock arithmetic.

Because the divisor is 7, the remainders must be less than or equal to 6.

So, the remainder are 0, 1, 2, 3, 4, 5, 6.

"0" corresponds to "Sunday"

"1" corresponds to "Monday"

"2" corresponds to "Tuesday"

"3" corresponds to "Wednesday"

"4" corresponds to "Thursday"

"5" corresponds to "Friday"

"6" corresponds to "Saturday"

How to solve problems on 7 day clock arithmetic ?  

Case 1 :

Let us say today is Friday. Find the day after 32 days from today.

"5" corresponds to "Friday"

If you want to find the day after 32 days from now, you have to add 32 to 5 and divide the result by 7. 

5 + 32  =  37

When 37 is divided by 7, the remainder is 2.

"2" corresponds to "Tuesday"

So, the day after 32 days from now is Tuesday.   

Case 2 :

Let us say today is Tuesday. Find the day after 4 days from today. 

"2" corresponds to "Tuesday"

If you want to find the day after 4 days from now, you have to add 4 to 2 and divide the result by 7. 

2 + 4  =  6   

Here 6 is less than 7. So, we can not divide 6 by 7. 

"6" corresponds to "Saturday"

So, the day after 4 days from now is Saturday.   

Case 3 :

Let us say today is Saturday. Find the day before 32 days from today.

"6" corresponds to "Saturday"

Subtract 32 from 6.

6 - 32  =  -26

Here, we get a negative value. 

In the case of getting a negative value, we have to find the next integer which is exactly divisible by 7.

That is, we have to get the next integer after 26 which is exactly divisible by 7.

It is 28.

Now, we have to write -26 in terms of -28.

So, -26 can be written as 

-26  =  -28 + 2

"2" corresponds to "Tuesday"

Therefore, the day 32 days before was Tuesday. 

Practice Problems

Problem 1 :

A work was assigned to John on a Tuesday. He took 72 days to complete this work. On what day did he complete the work ?

Solution :

Tuesday corresponds to 2 in 7 day clock arithmetic. 

We want to know the day which is 72 days after Tuesday. 

To get answer for our question, we have to do the following steps. 

Step 1 :

Add 72 to 2. 

72 + 2  =  74

Step 2 :

Divide 74 by 7.

74/7

Step 3 :

Take the remainder, when 74 is divided by 7.

The remainder is 4. 

"4" corresponds to "Thursday"

So, John completed the work on Thursday.

Problem 2 :

What will be the day 20 days after Monday ? 

Solution :

Monday corresponds to 1 in 7 day clock arithmetic. 

We want to know the day which is 20 days after Monday. 

To get answer for our question, we have to do the following steps. 

Step 1 :

Add 20 to 1.

1 + 20  =  21

Step 2 :

Divide 21 by 7.

21/7

Step 3 :

Take the remainder, when 21 is divided by 7.

The remainder is 0. 

"0" corresponds to "Sunday"

So, the day 20 days after Monday will be Sunday.

Problem 3 :

What was the day 23 days before Monday ? 

Solution :

Monday corresponds to 1 in 7 day clock arithmetic. 

We want to know the day which is 23 days before Monday. 

To get answer for our question, we have to do the following steps. 

Step 1 :

Subtract 23 from 1.

1 - 23  =  -22

Step 2 :

We get negative value in step 1 and also 22 is not divisible by 7.

So, find the next integer after 22 which is exactly divisible by 7.

That is 28. 

Step 3 :

Write -22 in terms of -28.

So, -22 can be written as

-22  =  -28 + 6

"6" corresponds to "Saturday" 

So, the day 23 days before Monday was Saturday.

Problem 4 :

Michael was assigned a work on Thursday. He agreed to complete the work after 25 days. But he took two more days to complete the work. On what day did he complete the work ?

Solution :

Thursday corresponds to 4 in 7 day clock arithmetic. 

We want to know the day which is (25+2=) 27 days after Thursday. 

To get answer for our question, we have to do the following steps. 

Step 1 :

Add 27 to 4.

27 + 4  =  31

Step 2 :

Divide 31 by 7. 

31/7

Step 3 :

Take the remainder, when 31 is divided by 7.

The remainder is 3. 

"3" corresponds to "Wednesday" 

So, Michael completed the work on Wednesday.

Problem 5 :

Mr. Lenin's birthday was on Monday. He visited Newyork 28 days before his birthday. On what day, did he visit Newyork ?

Solution :

Lenin's birthday was on Monday. Monday corresponds to 1 in 7 day clock arithmetic 

We want to know the day which was 28 days before Monday. 

To get answer for our question, we have to do the following steps. 

Step 1 :

Subtract 28 from 1.

1 - 28  =  -27 

Step 2 :

We get negative value in step 1 and 27 is not divisible by 7.

So, find the next integer after 27 which is exactly divisible by 7.

That is 28. 

Step 3 :

Write -27 in terms of -28.

So, -27 can be written as

-27  =  -28 + 1

"1" corresponds to "Monday" 

So, Lenin visited Newyork on Monday.

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