Questions based on Integral solutions and remainders based on factorial often feature in CAT, XAT and other MBA entrance exams. Such questions may belong to number system or algebra. Some of the typical questions asked from these areas are discussed below:

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**Question: How many integral solutions are possible for the equation |X| +|Y|..... = N?**

Looking at the complexity of the above equation and the number of cases it would involve, many of us would leave such a question in the exam.

But this question can be solved within seconds using a simple but hardly known trick.

In this article, we have shared the trick which will help you solve these type of questions within very less time.

Let us look at the solution for the above asked problem first i.e.

*How many integral solutions are possible for the equation |X| +|Y|..... = N?*

When we are asked to calculate the integral solutions possible for the equation |X| +|Y|..... = N, there are 3 very important cases that we need to consider.

**Case 1: **|X| +|Y|= N where N is a natural number

**Case 2:** |X| +|Y|+|Z|= N where N is a natural number

**Case 3:** |X| +|Y|+|Z|+|W|= N where N is a natural number

*Let us discuss these cases in detail.*

**Case 1: |X| +|Y|= N where N is a natural number**

In this case, total number of integral solutions will be=4N

**Example:** How many integral solutions are possible for the equation |X| +|Y|= 10

**Solution:** Total number of integral solutions =4*10=40

**Case 2: How many integral solutions are possible for the equation |X| +|Y|+|Z|= N where N is a natural number**

In this case, total number of integral solutions will be=4N^{2} + 2

**Example:** How many integral solutions are possible for the equation |X| +|Y|+ |Z|= 10

**Solution:** Total number of integral solutions =4*10^{2} +2 =402

**Case 3: How many integral solutions are possible for the equation |X| +|Y|+|Z|+|W|= N where N is a natural number**

In this case, total number of integral solutions will be=8N (N^{2} + 2)/3

**Example:** How many integral solutions are possible for the equation |X| +|Y|+ |Z|+|W|= 10

**Solution:** Total number of integral solutions =8*10*102/3 = 2720

*Another frequently asked question is related to remainders based on factorials. *The three type of questions are:

**Type 1: **What is the remainder when (P-1)! is divided by P where P is a prime number

**Type 2: **What is the remainder when (P-2)! is divided by P where P is a prime number

**Type 3: **What is the remainder when (P-3)! is divided by P where P is a prime number

**Type 1: When (P-1)! is divided by P where P is a prime number, the Remainder is (P-1).**

**Example:** If P=17 then (p-1)=16

When 16! Is divided by 17 the remainder is 16.

**Type 2: When (P-2)! is divided by P where P is a prime number, the Remainder is 1.**

**Example:** If P=17 then (p-2)=15

When 15! Is divided by 17 the remainder is 1

**Type 3: When (P-3)! is divided by P where P is a prime number, the Remainder is (P-1)/2.**

**Example:** If P=17 then (p-3)=14

When 14! Is divided by 17 the remainder is (17-1)/2=8

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