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Properties of square numbers

Here are some other properties of square numbers:

1. The number of zeros at the end of a perfect square ending with zeros is always even. e.g. 100, 400 etc.
2.  
3. No square number ends with 2,3, 7 or 8. e.g. 12, 13, 18 etc are not squares
4.  
5. A square leaves a remainder of 0 or 1 when it is divided by 3 i.e, if on dividing a number by 3, we get the remainder as 2, then the number is not a square. e.g. 8 divided by 3 leaves a remainder 2; its not a square.
6.  
7. When divided by 5, a square number leaves a remainder of either 0 or 1 or 4 i.e, if on dividing a number by 5, we get a remainder of 2 or 3, then the number is not a perfect square. e.g. 7, when divided by 5 leaves 2 as remainder – is not a square whereas 9 – a perfect square – when divided by 5 leaves 4 as remainder.
8.  
9. A square, when divided by 7, leaves a reminder of either 0 or 1 or 2. or 4. i.e, if on dividing a number by 7 we get a reminder as either 3 or 5 or 6, then the number is not a perfect square. e.g. 20, when divided by 7 leaves 6 as remainder and is not a square whereas 16, when divided by 7, leaves 2 as remainder.
10.  
11. If a number is even, then its square is also even.
12. If a number is odd, then its square is also odd.

Please keep in mind

Please note that it is not true other way round. for example, any number which leaves 0'0' (zero) as remainder after being divided by 3 need not be a square number e.g. 18 leaves 0 as remainder when divided by 3 but is not a square 

METHODS FOR FINDING SQUARE ROOTS

By Method Of Successive Subtraction :

We subtract the numbers 1, 3, 5, 7, 9, 11 …. successively till we get zero. The number of subtractions will give the square root of the number.

Ex:      Find the square root of 64 using the method of successive subtraction.

1.  64-1=63;
2. 63-3=60;
3. 60-5=55;
4. 55-7=48;
5. 48-9=39;
6. 39-11=28;
7. 28-13=15;
8. 15-15 = 0

The number of subtractions to yield zero is 8

By Method of Prime Factorization

Take the number (n) whose square root is required.

• Write all the prime factors of n.
• Pair the factors such that primes in each pair thus formed are equal
• Choose one prime from each pair and multiply all such primes.
• The product of these primes is the square root of n.

Ex:      225 = 3 x 3 x 5 x 5;
so the square root of 225 = 3 x 5 = 15

By Method of Division:

The method of finding square root by the prime factorization method is efficient only if the number has few prime factors. When the given number has more than five digits it is difficult to obtain prime factors. To overcome this difficulty, we use an alternative method called division method.

 Procedure:

1. Place a bar over every pair of digits starting from the units digit.
2. Find the largest number whose square is less than or equal to the number under   the bar to the extreme left.
3. Take this number as the divisor as well as the quotient and the number under the bar to the extreme left as the dividend. Divide and get the remainder.
4. Bring down the number under the next bar to the right of the remainder, this is the new dividend.
5. Double the quotient and enter it with a blank on the right for the next digit of the next possible divisor.
6. Guess the largest possible digit to fill the blank and also to become the new digit as the quotient. Now we get a remainder.
7. Bring down the number under the next bar to the new remainder.
8. Again repeat the above steps, till all bars have been considered. The final quotient is the square root of the given number.

 Ex:      Find the square root of 729

  2       7 29      27
+2      4
————-
47      329
           329
 ————-
             0

So, the square root of 729 is 27.

Please note that all the properties described in sections above apply in this case too. eg. last digit of 729 is 9 and its square root has 7 in unit place, when divided by 3, it leaves remainder of 0; when divided by 7, it leaves a remainder of 1.

Please check applicability of other properties too

Square Root Of Rational Numbers

Properties Of Square Roots:

 If X and Y are two positive numbers, then

          i) √X x √Y = √XY               ii) √(^X⁄_Y) = ^√X⁄_√Y
        iii) √X+Y ≠ √X + √Y          iv) √X-Y ≠ √X – √Y

Using these results, we can find the square roots of rational numbers.

      Find the value of √5¹⁄₁₆

            Sol:     
5¹⁄₁₆ = ⁸¹⁄₁₆

√⁸¹⁄₁₆ 

= ^√81⁄_√16 

= ⁹⁄₄ 

= 2¹⁄₄

Square Root Of A Decimal Number

Procedure:

1. Place bars on the integral part of the number in the usual manner
2. Now, Place bars on the decimal part on every pair of digits beginning with the first decimal place.
3. Find the square root by the division method as usual.

            Ex:

  2       0.000484       022
  0          00
    ————-
  2              04
  2              04
         ————-
  42                 84
                       84
            ————-
                        0