1. Find the smallest number which must be Added to 180 to make it perfect Square

Solution:

	1	3
1	1	80
1	1
23		80
3		69
26		11

 
Dividend = 180
Square Root = 13
Remainder = 11


Here, Quotient = 13, Remainder = 11

`:.` Number to be added

`= 2 × "Quotient" - "Remainder" + 1`

`= 2 × 13 - 11 + 1`

`= 16`

`:.` Perfect square number = 180 + 16 = 196

`:. sqrt(196) = 14`

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2. Find the smallest number which must be Added to 200 to make it perfect Square

Solution:

	1	4
1	2	00
	1
24	1	00
		96
28		4

 
Number = 200
Square Root = 14

Here, Quotient = 14, Remainder = 4

`:.` Number to be added

`= 2 xx "Quotient" - "Remainder" + 1`

`= 2 xx 14 - 4 + 1`

`=25`

`:.` Perfect square number `=200 + 25=225`

`:. sqrt(225)=15`

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3. Find the smallest number which must be Substracted from 200 to make it perfect Square

Solution:

	1	4
1	2	00
	1
24	1	00
		96
28		4

 
Number = 200
Square Root = 14
`:.` Number to be substracted = 4

`:.` Perfect square number = 200 - 4=196

`:. sqrt(196)=14`

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4. Find the smallest number which must be Multiplied to 200 to make it perfect Square

Solution:
First find factors of 200

2	200
2	100
2	50
5	25
5	5
	1

`200=2 xx 2 xx 2 xx 5 xx 5`

`200=2^3 * 5^2`

We know that if a number is to be a perfect square then each of its prime factors must occur twice.
The smallest number by which the given number must be multiplied in order that the Result is a perfect square is `2=2`

Result `= (2^3 * 5^2) * (2)`

`=2^4 * 5^2`

`=400`

`sqrt(400)=sqrt(2^4 * 5^2)` (Taking squareroot of the Result)

`=2^2 * 5`

`=20`

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5. Find the smallest number which must be Divided from 200 to make it perfect Square

Solution:
First find factors of 200

2	200
2	100
2	50
5	25
5	5
	1

`200=2 xx 2 xx 2 xx 5 xx 5`

`200=2^3 * 5^2`

We know that if a number is to be a perfect square then each of its prime factors must occur twice.
The smallest number by which the given number must be divided in order that the Result is a perfect square is `2=2`

Result `= (2^3 * 5^2) / (2)`

`=2^2 * 5^2`

`=100`

`sqrt(100)=sqrt(2^2 * 5^2)` (Taking squareroot of the Result)

`=2 * 5`

`=10`

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