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Properties of

Perfect Squares

riends,  we  have  alreadyposted  3  shortcuttechniques  for  findingSquares.  In  this post we

shall  discuss  some  basicproperties of Perfect Squares.

1. A  number  ending  in  2,  3,  7or  8  is  never  a  perfectsquare.  or  all  the  squarenumbers end with 0, 1, 4, 5,6 or 9 at units place. We canverify  this  statement  by

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observing  the  squares  tablehere. 

Note : The converse ofthe  above  statement  isnot  true.  i.e.,  if  anumber ends in 0, 1, 4,5, 6 or 9, then it  is notnecessarily  a  squarenumber.  For  example,170, 251, 3584, etc arenot  Square  numbers,though  these  end  with0, 1, 4.

2. If a number has 1 or 9 in theunit's  place,  then  its  squareends  in  1.  This  statementcan be verified by observingthe squares table.

3. When a square number endsin  6,  the  number,  whosesquare  it  is, will have either4 or 6 in the unit's place. 

4. The  number  of  zeros  at  theend  of  a  perfect  square  isalways even.

For Example : 10000 =

1002,  2500  =  502,

490000=7002

5. Squares  of  even  numbersare always even and squaresof  odd  numbers  are  alwaysodd.

For  Example  :  22  =  4,

82=64,  402=1600,

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52=25,  92=81,

172=289

6. For  any  two  consecutivenatural  numbers  n  and(n+1), we have

(n+1)  2n2      =   (n+1+n)(n+1n)      =   (n+1)+n

 For Example :

11  2  –  10  2  =11+10 = 21

15 214 2 = 15+14= 29

19 218 2 = 19+18= 37 etc

7.   A  triplet  (x,  y,  z)  of  threenatural numbers x, y and z iscalled  a  Pythagorean  triplet

if X2  + y2  = z2    

for example,  (6, 8, 10)is a Pythagorean triplet.

Since 62 + 82 =  36+64

= 100  and 102 = 100

Note  :  For  any  naturalnumber  n  greater  than1,  the  Pythagoreantriplet  is  given  by  (2n,

n21, n2+1)

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n21, n2+1)

Example  :  Findthe  other  twonumbers  of  aPythagorean triplet,one  number  ofwhich is 12.

Solution :

For  anynaturalnumber m  thePythagoreanTriplet  =  2m,

m21, m2+1 

let m = 6

So,  2m =12

m21  =

621  =361  =35

m2+1  =

62+1  =36+1  =37

So,  the  othertwo  numbersof  thePythagoreanTriplet  are  35and 37.

8.   The  square  of  a  naturalnumber  m  is  equal  to  thesum  of  the  first  m  odd

numbers. 

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numbers. 

Thus  12  =1  =  sum  ofthe first 1 odd number

22 = 4 = 1+3 = sum ofthe first 2 odd numbers

32 =9 = 1+3+5 = sumof  the  first  3  oddnumbers 

52 =25 = 1+3+5+7+9= sum of the first 5 oddnumbers an so on.

9.  We can express the squareof  any  odd  number  as  thesum  of  two  consecutivepositive integers.

For example, 

52 = 25 = 12+13

112  =  121  =60+61

412  =  1681  =840+841 etc 

 Note : The converse ofthe  above  statement  isnot true. i.e., the sum ofany  two  consecutivepositive  integers  is  notnecessarily  a  perfectsquare  or  squarenumber.

Ex  :  14+15  =  29,which  is  not  asquare number.

10. For  any  natural  number  n

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10. For  any  natural  number  ngreater  than 1,  (n+1) x  (n

1) = n21

Using  this  property,  wecan  find  the  product  oftwo consecutive even orodd  natural  numberseasily.

7  x  9  =  (81)  x

(8+1) = 63 =  821

15 x 17 = (161) x(16+1)    =  255

=1621

 24 x 26 = (251)x (25+1) = 624 =

2521

11. Observe the following :

  12  =  1  and  22=4,

Between  12  =  1  and

22=4,  the  numbers  are2 3.i.e., there are 2 x 1=  2  non  squarenumbers.

  22  =  4  and  32  =  9,

Between  22 = 4 and 32

= 9, the numbers are 5,6, 7, 8 i.e., there are 2x  2  =  4  non  squarenumbers.

32  =  9  and  42    =  16,

Between 32 = 9 and 42

 = 16, the numbers are

10, 11, 12, 13, 14, 15.

10, 11, 12, 13, 14, 15.i.e., there are 2 x 3 = 6non square numbers.

82  =  64  and  92=81,

Between  82  =  64  and

92=81,  the  numbersare 65, 66, 67, 68, 69,70, 71, 72, 73, 74, 75,76, 77, 78, 79, 80. i.e.,there  are  2  x  8  =  16non square numbers.

Thus, We can  say  thatthre  are  2n  nonperfect  squarenumbers  between  thesquares of the numbersn and (n+1).

12. Study the following pattern :

22 = 4 = 3x1+1

or  22  =  4  =4x1 

 32 = 9 = 3x3

or  32  =  9  =4x2+1

 42= 16 = 3x5+1

or  42  =  16  =4x4

52 = 25 x 3x8+1

52=  25  =4x6+1

62 = 36 = 3x12

Or 32 = 36 =

Or 32 = 36 =4x9

    From  the  above  wecan say that

Squares  ofnumbers  (greaterthan  1)  can  bewritten as multiplesof 3 or multiples of3 plus 1.

Squares  ofnumbers  (greaterthan 1) can also bewritten as multiplesof 4 or multiples of4 plus 1.

 This property is very useful whenwe  want  to  check  whether  anumber  is  a  perfect  square  ornot.  For  example,  if we  divide  anumber  by  3,  and  get  theremainder 2, then the number  isnot a perfect square.

Read  shortcut  techniques  forfinding squares from here

Shortcut  Technique  forfinding squares 1

Shortcut  Technique  forfinding Squares 2

Shortcut  Technique  forsquares pdf download

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• Reply •

swathi  •  2 years ago

in the above material number10 para there should be 7x9not 7x8

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Anonymous•  2 years ago

good post

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Shiv RK  •  2 years ago

Typing mistake friend.

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Typing mistake friend.Corrected now.... Thanks forthe update :)

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Anonymous•  2 years ago

IBPS CWE SPL III Notificationout

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Radhika  •  2 years ago

IBPS CWE SPL III Notiifcationcame

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Anonymous•  2 years ago

PLS EXPLAIN PARA 10

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Deb Chakraborty•  4 months ago

5,8,9,11

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