Perfect Square Number & Applications-Number System

 A number made by squaring a whole number is called a perfect square number. In other words, it is the product of some (natural number/0) with itself.

Test by looking at last/unit digit:-

In decimal number system, any number is formed by 0,1,2,3,4,5,6,7,8,9.
0² = 0
1² = 1
2² = 4
3² = 9
4² = 16
5² = 25
6² = 36
7² = 49
8² = 64
9² = 81

By observing the unit digits only we see every perfect square number ends in 0,1,4,5,6&9 only.

Test by looking at the last two digits (Best concept):-

Every number can be written as (50n ± x), where x is a number from 0 to 25 and n = 0,1,2,3…so on (any whole number).

 Here by 50n, I just mean any multiple of 50 i.e. 0, 50, 100, 150, ……

0 to 25 = (0 to 25) itself.

25 to 50 = 50 – (25 to 0)

50 to 75 = 50 + (0 to 25)
75 to 100 = 100 – (25 to 0)
100 to 125 = 100 + (0 to 25)
125 to 150 = 150 – (25 to 0)
and so on. 

(50n ± x)² = 2500n² ± 100nx + x² = 100 [25n² ± nx] + x²

The last two digits of each of 2500n² and 100nx  will be 00. Thus the last two digits of the RHS, and hence of the LHS, will be the last two digits of x².

And since x is a number from 0 to 25, you should know the last two digits of x².

Because of the above logic, the ONLY possible values for the last two digit of any perfect square number is the last two digits of squares of 0 to 25:-

00, 01, 04, 09, 16, 25, 36, 49, 64, 81, 00, 21, 44, 69, 96, 25, 56, 89, 24, 61, 00, 41, 84, 29, 76 and 25.
Clearly, the unique last two digits occur in squares of 1 to 24 as 25 exists among them.

Test by Digit Sum (Remainder on dividing by 9 or simply add up all the digits till it becomes Single digit) :-

In decimal number system, any number is formed by 0,1,2,3,4,5,6,7,8,9. Remember that digit sum should be a single digit 0/1/2/3/4/5/6/7/8 as it is the remainder obtained on dividing by 9.
0² = 0
1² = 1
2² = 4
3² = 9 = 0
4² = 16 = 1+6 =7
5² = 25 = 2+5 =7
6² = 36 = 3+6=9=0
7² = 49 = 4 (9 itself is divisible by 9 so it can be neglected.)
8² = 64 =6+4 =10=1 (0 doesn't make sense in addition)
9² = 81 = 8+1=9=0

As You observed above perfect square number should have digit sum 0/1/4/7 only.

# Necessary conditions for any number to be a perfect square number:-

1. If any number is perfect square no. then it ends in 0/1/4/5/6/9. But the converse is not necessarily true that if the last digit of a no. is 0/1/4/5/6/9 then it must be a perfect no.[It may or may not]

2. So, obviously any number ending in 2/3/7/8 will not be a perfect square number.

3.Last 2 digits of any perfect square number will be the same as the last 2 digits in squares of 1 to 24.

4. If a number is a perfect square no. then its digit sum should be 0/1/4/7. But the converse is not necessarily true that if the digit sum of a no. is 0/1/4/7 then it will be a perfect no.

Q. N is a 3 digit number & N² = .... 94 . How many values of N are possible?

1. 0   2. 1  3. Infinitely many  4. None of These 

Sol:-  Last 2 digits of any perfect square number will be the same as the last 2 digits in squares of 1 to 24.

But 94 doesn't come in the square of any number from 1 to 24.

So, the possible value is 0 & hence option 1. is correct.

Q. The 10th place digit of a perfect square no. is 7 then how many possible values for unit digit?

Sol:- Let the number be _ _ _ 7?

Firstly, a perfect square number must end in 0/1/4/5/6/9.

Simultaneously, only 76 is the last 2 digits occurring in the square from 1 to 24. (24² = 576)

Therefore the possible value is 1.

Q. AB5N is a perfect square no. How many values of N are possible?

Sol:- Last 2 digits of any perfect square number will be the same as the last 2 digits in squares of 1 to 24.

A perfect square number must end in 0/1/4/5/6/9.

But 56 is the only possible in the square of any number from 1 to 24. (16² = 256)

Q. The square root of which of the following is a rational no. ?

1. 6250.49  2. 1250.49  3. 1354.24  4. 5768.28

Sol:-  Don't get confused with the trick of the question. It is asking to find out the perfect square number as square root is a rational no.

Also, decimal point (.) occurs leaving 2 places from RHS , hence denominator is 100 i.e. a  perfect square no. Now, we have to consider these options avoiding decimal point.

Clearly, 28 doesn't come in the squares od 1 to 24 hence we cancel option 4.

 Now, use digit sum method & we find as below:-

1. 625049 = 8  2. 125049 =3  3. 1354.24=1

Obviously, a perfect square number should have digit sum 0/1/4/7 only. Here Option 3. is correct,therefore.