## Blog

# If p be a prime number, p>3 and let x be the product of positive number 1, 2, 3,…,(p-1), then consider the following statements?(CAT, 2013)

- May 14, 2018
- Posted by: allexamshelps
- Category: CAT

No Comments

1,353 total views, 1 views today

**If p be a prime number, p>3 and let x be the product of positive number 1, 2, 3,…,(p-1), then consider the following statements**

**1. x is a composite number divisible by p.**

**2. x is a composite number not divisible by but some prime number greater than p may divide x.**

**3.x is not divisible by any prime number (p-2).**

**4. all prime numbers less than (p-1) divide x.**

**Which of the following statement is/are correct** **[CAT, 2012]**

#### **(a) I and II are correct **

**(b) II and III are correct**

**(c) III and IV are correct **

**(d) IV alone is correct**

**Solution:**

**If p be a prime number, p>3 and let x be the product of positive number:**

#### **The prime numbers greater than 3 are 5, 7, 11...**

**For example, we say that the prime number > 3 is 7**

**In this way, we take the positive numbers till 6 i.e. p-1 or 7-1=6**

** And calculate the product of p-1 positive numbers**

**1 x 2x 3x 4x 5 x 6= 720 i.e x=720**

**Now check the condition one by one.**

#### **1).** **x is a composite number divisible by p.**

**p in our case is 7 or p=7**

**\frac { 720 }{ 7 }**

**Not divisible, hence the first condition is false.**

#### **2).** **x is a composite number not divisible by but some prime number greater than p may divide x.**

**Prime numbers greater than 7 are 11, 13, 17, 19, 23, 27…**

**\frac { 720 }{ 11}**

**\frac { 720 }{ 13 }**

**\frac { 720 }{ 17 }**

** .**

** .**

** .**

**\frac { 720 }{ 27 }**

**Not divisible, hence the second condition is also false.**

**3). x is not divisible by any prime number (p-2).**

**(P-2) is 7-2=5 we have to choose the prime numbers till 5 only**

**The prime number starts from 2 and the next prime numbers are 3, 5, 7, 11 etc.**

**\frac { 720 }{ 2 } = 360**

**\frac { 720 }{ 3 } = 260**

**\frac { 720 }{ 5 } = 124**

**Divisible by all p-2 prime numbers, hence the third condition is also false.**

**4).all prime numbers less than (p-1) divide x.**

**(p-1) = 7-1=6**

**Prime numbers less than 6 are 2, 3, 5**

**\frac { 720 }{ 2 } = 360**

**\frac { 720 }{ 3 } = 260**

**\frac { 720 }{ 5 } = 124**

**So, all prime number less than p-1 divide x=720.**

#### ** ****Hence the option (d) is correct.**

** (Please leave your comments, to encourage us, to let us know how good/bad we are doing it (Please support your comments with reasons)).**

### Leave a Reply Cancel reply

1,354 total views, 2 views today