Perfect-Square

Various properties of perfect square numbers have been summarised as follows. If you learn how to utilise these properties to solve questions effectively, you would get an edge on your competitors.


1. Last digit of perfect square must be 0, 1, 4, 5, 6, 9

As you know a number can end any of 0-9 digits. Unit digit of a square number depends on unit digit of number of which square is being done. You can notice in the below table that square of 0-9 digits ends with 0, 1, 4, 5, 6, 9.

Number

1

2

3

4

5

6

7

8

9

0

Square

1

4

9

16

25

36

49

64

81

0

Possible unit digit
 of a square number

1

4

9

6

5

6

9

4

1

0


2. A number ending with 2, 3, 7, 8 can’t be perfect square. (Vice versa of statement 1)

3. Every zero in the end of a number causes 00 in the end of its square. See some examples in below table.

Number

10

40

100

500

10000

Square

100

1600

10000

250000

100000000


So if a number has ZEROs at unit place, then square number would end with even number of ZEROs.

4. A number ending with odd number of zeroes can’t be perfect square e.g. 9000, 70, 16000, can’t be perfect square.
Point 4 is just an extension of point 3.

5. A perfect square is either exactly divisible by 3 or leaves a remainder of 1 when divided by 3. That means, every perfect square can be represented in the form of 3n or 3n+1

Pick some square numbers randomly and check if it can be represented in the form of 3n or 3n+1. e.g. 

Number

3

4

12

13

31

37

101

Square

9

16

144

169

961

1369

10201

3n or (3n+1) form

3X3

3X5+1

3X48

3X56+1

3X320+1

3X456+1

3X3400+1


6. A perfect square is either exactly divisible by 4 or leaves a remainder of 1 when divided by 4. That means, every perfect square can be represented in the form of 4n or 4n+1

Pick some square numbers randomly and check if it can be represented in the form of 4n or 4n+1, e.g.

Number

3

4

12

13

31

37

101

Square

9

16

144

169

961

1369

10201

4n or (4n+1) form

4X2+1

4X4

4X36

4X42+1

4X240+1

4X342+1

4X2550+1


7. Digit sum of every square number will be either of 1, 4, 7, 9 but vice versa is not true. That means if a number has digit sum either of 1, 4, 7, 9, it is NOT NECESSARILY a perfect square.

Pick some numbers randomly and check it. e.g. 

Number

3

4

12

13

31

37

101

Square

9

16

144

169

961

1369

10201

Digit sum

9

1+6=7

9

1+6+9=16;

1+6=7

7

1

4



8. A perfect square ending with an odd digit, the ten’s digit must be an even digit.

Number

5

13

31

37

101

Square

25

169

961

1369

10201

10’s Digit

2(Even)

6 (Even)

6 (Even)

6 (Even)

0 (Even)


9. If a perfect square ends with 5, then ten’s digit must be 2. e.g. perfect square ends with 25 if unit digit is 5. 

Number

5

15

35

95

105

Square

25

225

1225

9025

11025

Last 2 digits

25

25

25

25

25


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  Posted on Tuesday, March 10th, 2015 at 1:54 PM under   Reasoning