In modular arithmetic, because there are 12 months in a year, it is the same as the 12 hour clock arithmetic .

So, in month arithmetic, the divisor is 12. Because the divisor is 12, the remainders must be less than or equal to 11.

Then, the remainder are 0,1,2,3,4,5,6,7,8,9,10,11

"0" corresponds to the "December"

"1" corresponds to "January"

"2" corresponds to "February"

"3" corresponds to "March"

"4" corresponds to "April"

"5" corresponds to "May"

"6" corresponds to "June"

"7" corresponds to "July"

"8" corresponds to "August"

"9" corresponds to "September"

"10" corresponds to "October"

"11" corresponds to "November"

**
**

**Case 1 :**

Let us say this month is March. Find the month after 32 months from now.

"3" corresponds to "March"

If you want to find the month after 32 months from now, you have to add 32 to 3 and divide the result by 12.

3 + 32 = 35

When 35 is divided by 12, the remainder is 11.

"11" corresponds to "November"

So, the month after 32 months from this month (March) is November.

**Case 2 :**

Let us say today is February. Find the month after 9 months from this month.

"2" corresponds to "February"

If you want to find the month after 9 months from this month, you have to add 9 to 2 and divide the result by 12.

2 + 9 = 11

Here 11 is less than 12. So, we can not divide 11 by 12.

"11" corresponds to "November"

So, the month after 9 months from this month (February) is November.

**Case 3 :**

Let us say this month is May. Find the month before 32 months from this month.

"5" corresponds to "May"

Subtract 32 from 5.

5 - 32 = -27

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 12.

That is, we have to get the next integer after 27 which is exactly divisible by 12.

It is 36.

Now, we have to write -27 in terms of -36.

So, -27 can be written as

-27 = -36 + 9

"9" corresponds to "September"

Therefore, the month before 32 was September.

**Problem 1 :**

What month is 19 months after July ?

**Solution :**

July corresponds to 7 in month arithmetic.

We have to find the month which is 19 months after July.

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

**Step 1 :**

Add 19 to 7.

7 + 19 = 26

**Step 2 :**

Divide 26 by 12.

26/12

**Step 3 :**

Take the remainder. when 26 is divided by 12.

The remainder is 2.

"2" corresponds to "February"

So, the month which is after 19 months will be February .

**Problem 2 :**

What month was 19 months before July ?

**Solution :**

July corresponds to 7 in month arithmetic.

We have to find the month which is 19 months before July.

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

**Step 1 :**

**Subtract 19 from 7.**

**7 - 19 = -12**

**Step 2 :**

**We get negative value in step 1.**

**But, 12 is exactly divisible by 12. **

**So, the remainder is 0. **

**Step 3 :**

"0" corresponds to "December"

**So, the month 19 months before July was December.**

**Problem 3 :**

Anderson took a work in the month of September and agreed to completed the work after 17 months. In which month, did he complete the work ?

**Solution :**

September corresponds to 9 in month arithmetic.

We want to find the month which is 17 months after September.

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

**Step 1 :**

**Add 17 to 9. **

**9 + 17 = 26**

**Step 2 :**

**Divide 26 by 12.**

**26/12**

**Step 3 :**

Take the remainder, when 26 is divided by 12.

The remainder is 2.

**"2" corresponds to "February" **

**So, **Anderson completed the work in February.

**Problem 4 :**

Jack started a work at in the month of June. He was given 38 months time to complete the work. But he took 3 more months to complete the work. In which month did he complete the work ?

**Solution :**

June corresponds to 6 in month arithmetic.

We want to find the month which is (38+3=) 41 months after June.

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

**Step 1 :**

**Add 41 to 6.**

**6 + 41 = 47**

**Step 2 :**

**Divide 47 by 12.**

**47/12**

**Step 3 :**

Take the remainder, when 47 is divided by 12.

The remainder is 11.

**"11" corresponds to "November" **

**So, Jack** completed the work in the month of November.

**Problem 5 :**

Now the month is March. Johnson was in japan 45 months before. In which month, was Johnson in Japan ?

**Solution :**

March corresponds to 3 in month arithmetic.

We want to find the month which was 45 months before March.

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

**Step 1 :**

**Subtract 45 from 3.**

**3 - 45 = -42**

**Step 2 :**

**We get negative value in step 1. **

**And also, 42 is not divisible by 12.**

**So, find the next integer after 42 which is exactly divisible by 12****.**

**That is 48.**

**Step 3 :**

Write -42 in terms of -48.

So, -42 can be written as

-42 = -48 + 6

**"6" corresponds to "June" **

**So, **Johnson was in Japan in the month of June.

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