### Perfect Square

1. A number is called a

**perfect square**if it is expressed as the square of a number.
2. E.g. 1, 4, 9, 16, 25, ... are called perfect squares (1x1 = 1, 2x2 = 4, 3 x 3 = 9...)

3. In square numbers, the digits at the unit’s place are always 0, 1, 4, 5, 6 or 9.

4. The numbers having 2, 3, 7 or 8 at its units' place are not perfect square numbers.

**Q1: Which one of the following number is a perfect square:**

a) 622

b) 393

c) 5778

d) 625

**Answer**: d.

5. If a number ends with odd number of zeros then it is not a perfect square.

**Q2: Check which of the following is a not a perfect square.**

a) 81000

b) 8100

c) 900

d) 6250000

**Answer**: a) 81000 (= 9

^{2}x 10

^{2}x 10)

6. The square of an even number is an even number while the square of an odd number is an odd number.

7. If n is a positive whole number then

**(n+1)**

^{2}- n^{2}= 2n + 1or 2n numbers in between the squares of the numbers n and (n + 1)

**Q3: Which of the following perfect square numbers, is the square of an odd number?**

**289, 400, 900, 1600**

a) 289

b) 400

c) 900

d) 1600

**Answer**: a) 289

**Q5: Which of the following perfect square numbers, is the square of a even number?**

**361, 625, 4096, 65536**

a) 361

b) 625

c) 4096

d) 2601

**Answer**: c)

**Q6: How many natural numbers lie between squares of 12 and 13.**

a) 22

b) 23

c) 24

d) 25

**Answer**: c) 24 (2x12 = 24)

**Q7: A yoga instructor wants to arrange maximum possible number of 6000 students in a ground so that the number of rows is same as the number of columns. How many rows will be there if 71 students were left out after the arrangement.**

a) 80

b) 88

c) 77

d) 78

**Answer:**c) 77. (Hint: Remaining students = 6000 - 71 = 5929 = 11 x 11 x 7 x 7 = 11 x 7 = 77)

**Q8: The perfect square number between 30 and 40 is:**

a) 32

b) 35

c) 36

d) 39

**Answer:**c) 36

**Q9: Can a prime number be a perfect square?**

a) True

b) False

**Answer:**b) False

8. Unit digits of x

^{n}where x, n ∈ W

Units Digit of Number (x) | Units Digit of the number (x ^{n}) | No. of possibilities |
---|---|---|

0 | 0 | 1 |

1 | 1 | 1 |

2 | 2,4,6,8 | 4 |

3 | 3,9,7,1 | 4 |

4 | 4,6 | 2 |

5 | 5 | 1 |

6 | 6 | 1 |

7 | 7,9,3,1 | 4 |

8 | 8,4,2,6 | 4 |

9 | 9,1 | 2 |

**Q10: Find the units digit of (564)**

^{64}.
a. 2

b. 4

c. 6

d. 8

**Answer:**(c) 6.

Hint: Option a and d can be easily eliminated, since 2 and 8 never comes in units place in 4

^{x}If you see pattern (4 x 4 = 4^{2}= 16 i.e. 4^{2x}= 6 at units place) and (4 x 4 x 4 = 4^{3}= 64 i.e. 4^{2x+1}= 4 at units place). Similarly, (4)^{64}= 6.9. The sum of first m odd natural numbers is a perfect square and is equal to m

^{2}

10. 1

^{2}+ 2

^{2}+ 3

^{2}+ ... + n

^{2}= n(n+1)(2n+1)/6

**Q11. 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 is a perfect square of number?**

a) 8

b) 7

c) 6

d) 9

**Answer:**a) 8

**Q12: The square of an integer is called a perfect square number. If x is a perfect square number, then its next one is**

a. x+1,

b. x

^{2}+1,

c. x

^{2}+2x+1,

d. x+2 √x+1.

**Answer**: d. (Hint: if x is perfect square, then number is √x. Next perfect square will be (√x + 1)

^{2}

i.e. x+2
√x+1.

**Q12: As shown in figure below, the area of three squares are given. Find the perimeter of ΔABC**.

a. 12 units

b. 12.5 units

c. 19.5 units

d. 20 units

**Answer**: a. (Hint: Length of each side of square is √25 = 5, √9 = 3 and √16 = 4. Perimeter = 5+3+4 = 12units)

11. The sum of 1 and the product of any four consecutive integers is a perfect square.

E.g. 3 x 4 x 5 x 6 + 1 = 360 + 1 = 361 = √361 = 19

12. To find square root of a number, we generally use following two methods:

- Factorization Method: Finding prime factors of a number
- Long Division method.

**Q13: Find the least perfect square number which is divisible 16, 20 and 24?**

**Answer**: Take the LCM of 16, 20 and 24 which is the least number divisible by all three.

i.e. LCM(16,20,24) = 4x4x5x3 = 240

To make 240 perfect square (multiply by 3x5) = 4

^{2}x 5

^{2}x 3

^{2}= 3600 ...(answer)

**______**

**Q14. Compute √0.0016**

Answer:

**√**(0.0016) =

**√**(16/10000) =

**√**(4

^{2}x 10

^{-4}) = 4 x 10

^{-2}= 0.04

**Q15: Prove that product of four consecutive integers plus one is a perfect square number.**

Answer: Let the four consecutive integers are: x, x+1, x+ 2 and x+3.

Let P is the product.

⇒ P = x(x+1)(x+2)(x+3)

P = (x

^{2}+ x)(x+2)(x+3) = (x

^{3}+ 3x

^{2}+ 2x)(x+3) = x

^{4}+ 6x

^{3}+ 11x

^{2}+ 6x

Adding 1 to both sides,

P + 1 = x

^{4}+ 6x

^{3}+ 11x

^{2}+ 6x + 1 = (x

^{2}+ 3x + 1)

^{2}⇒ is a perfect square.

**Q16: If (17)**

^{2}is subtracted from a square of a number (x), the result obtained is 1232. Find the number 'x'.Answer: as per the question, x

^{2}- 17

^{2}= 1232

⇒ x

^{2}= 1232 + 17

^{2}= 1232 + 289 = 1521

⇒ x =

**√**(1521) =

**39**

How to solve root x - one by root x when x= 9+4root5

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