*Wednesday, May 16th, 2018*

Figuring out the last digit is the same as figuring out the remainder of a number when divided by 10, but I guess you already know that. Figuring out the last two digits is the same as figuring out the remainder of a number when divided by 100. However, if you wish to figure the remainder when the divisor is not 10 or 100, I suggest you read on.

Basic idea of remainders can be used to solve complicated problems.

There is nothing special or unique about this idea. At first glance it seems like something really obvious. But it is its usage that makes it special and helpful in questions related to remainders.

Let us look at couple of examples to see how this can be used effectively. In the first example we will see the idea that will work in cases of a^{b} and in the second example we will see the idea that will work in case of

**Example 1:** Find out the remainder when 25^{25} is divided by 7.

**Example 2:** Find out the remainder when Â is divided by 7.

We know that 4^{3} gives a remainder of 1, when divided by 7.

So, if we have 4^{3k}, it will give a remainder of 1 when divided by 7

If we have 4^{3k+1}, it will give a remainder of 4 when divided by 7

If we have 4^{3k+2}, it will give a remainder of 2 when divided by 7.

So, we need to reduce 26^{27} as 3k or 3k+1 or 3k+2. If we can do that, we will know the answer. So our

task has now been reduced to figuring out

**Note:** As you can see in solving this example, we have used the concept of negative remainder. In some cases, using the negative remainder can reduce your calculations significantly. It is recommended that you practice some questions using negative remainders instead of positive ones.

While trying to find out the remainder, if the dividend (M) and the divisor (N) have a factor (k) in common; then

- Â Cancel out the common factor
- Â Find out the remainder from the remaining fraction
- Â Multiply the resulting remainder with the common factor to get the actual remainder

*In equation format, this can be written as:*

**Example:** Find out the remainder when 4^{15} is divided by 28.

While trying to find out the remainder, if the divisor can be broken down into smaller co-prime factors; then

**Note:** If you wish to read more about it and how it happens, I suggest you read about the *Chinese Remainder Theorem**. *

*Example: Find out the remainder when 7 ^{15} is divided by 15.*

*By using the above fundas, solving remainder problems will get a little easier. But if you are thinking, that this is all you need to know to solve remainder problems in CAT â€“ I beg to differ. Great mathematicians like Euler, Fermat & Wilson developed some theorems that come in handy while solving remainder questions. *

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