sponsored links
 A number ending in 2, 3, 7 or 8 is never a perfect square. or all the square numbers end with 0, 1, 4, 5, 6 or 9 at units place. We can verify this statement by observing the squares table here.
 Note : The converse of the above statement is not true. i.e., if a number ends in 0, 1, 4, 5, 6 or 9, then it is not necessarily a square number. For example, 170, 251, 3584, etc are not Square numbers, though these end with 0, 1, 4.
 If a number has 1 or 9 in the unit's place, then its square ends in 1. This statement can be verified by observing the squares table.
 When a square number ends in 6, the number, whose square it is, will have either 4 or 6 in the unit's place.
 The number of zeros at the end of a perfect square is always even.
 For Example : 10000 = 100^{2}, 2500 = 50^{2}, 490000=700^{2 }
 Squares of even numbers are always even and squares of odd numbers are always odd.
 For Example : 2^{2} = 4, 8^{2}=64, 40^{2}=1600, 5^{2}=25, 9^{2}=81, 17^{2}=289
 For any two consecutive natural numbers n and (n+1), we have
 (n+1)^{ 2}n^{2 } = (n+1+n)(n+1n) = (n+1)+n
 For Example :

11^{ 2} – 10^{ 2} = 11+10 = 2115^{ 2}14^{ 2} = 15+14 = 2919^{ 2}18^{ 2} = 19+18 = 37 etc
 A triplet (x, y, z) of three natural numbers x, y and z is called a Pythagorean triplet if X^{2} + y^{2} = z^{2}
 for example, (6, 8, 10) is a Pythagorean triplet.
 Since 6^{2} + 8^{2} = 36+64 = 100 and 10^{2} = 100
 Note : For any natural number n greater than 1, the Pythagorean triplet is given by (2n, n^{2}1, n^{2}+1)
 Example : Find the other two numbers of a Pythagorean triplet, one number of which is 12.
 Solution :
 For any natural number m the Pythagorean Triplet = 2m, m^{2}1, m^{2}+1
 let m = 6
 So, 2m = 12
 m^{2}1 = 6^{2}1 = 361 = 35
 m^{2}+1 = 6^{2}+1 = 36+1 = 37
 So, the other two numbers of the Pythagorean Triplet are 35 and 37.
 The square of a natural number m is equal to the sum of the first m odd numbers.
 Thus 1^{2} =1 = sum of the first 1 odd number
 2^{2} = 4 = 1+3 = sum of the first 2 odd numbers

3^{2} =9 = 1+3+5 = sum of the first 3 odd numbers

5^{2} =25 = 1+3+5+7+9 = sum of the first 5 odd numbers an so on.
 We can express the square of any odd number as the sum of two consecutive positive integers.
 For example,
 5^{2} = 25 = 12+13
 11^{2} = 121 = 60+61
 41^{2} = 1681 = 840+841 etc
 Note : The converse of the above statement is not true. i.e., the sum of any two consecutive positive integers is not necessarily a perfect square or square number.
 Ex : 14+15 = 29, which is not a square number.
 For any natural number n greater than 1, (n+1) x (n1) = n^{2}1
 Using this property, we can find the product of two consecutive even or odd natural numbers easily.
 7 x 9 = (81) x (8+1) = 63 = 8^{2}1
 15 x 17 = (161) x (16+1) = 255 =16^{2}1
 24 x 26 = (251) x (25+1) = 624 = 25^{2}1
 Observe the following :
 1^{2} = 1 and 2^{2}=4, Between 1^{2} = 1 and 2^{2}=4, the numbers are 2 3.i.e., there are 2 x 1 = 2 non square numbers.
 2^{2} = 4 and 3^{2} = 9, Between 2^{2} = 4 and 3^{2} = 9, the numbers are 5, 6, 7, 8 i.e., there are 2 x 2 = 4 non square numbers.
 3^{2} = 9 and 4^{2} = 16, Between 3^{2} = 9 and 4^{2} = 16, the numbers are 10, 11, 12, 13, 14, 15. i.e., there are 2 x 3 = 6 non square numbers.
 8^{2} = 64 and 9^{2}=81, Between 8^{2} = 64 and 9^{2}=81, the numbers are 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80. i.e., there are 2 x 8 = 16 non square numbers.
 Thus, We can say that thre are 2n non perfect square numbers between the squares of the numbers n and (n+1).
 Study the following pattern :
 2^{2} = 4 = 3x1+1
 or 2^{2} = 4 = 4x1
 3^{2} = 9 = 3x3
 or 3^{2} = 9 = 4x2+1
 4^{2}= 16 = 3x5+1
 or 4^{2} = 16 = 4x4
 5^{2} = 25 x 3x8+1
 5^{2}= 25 = 4x6+1
 6^{2} = 36 = 3x12
 Or 32 = 36 = 4x9
 From the above we can say that
 Squares of numbers (greater than 1) can be written as multiples of 3 or multiples of 3 plus 1.
 Squares of numbers (greater than 1) can also be written as multiples of 4 or multiples of 4 plus 1.
This property is very useful when we want to check whether a number is a perfect square or not. For example, if we divide a number by 3, and get the remainder 2, then the number is not a perfect square.
Read shortcut techniques for finding squares from here
sponsored links
in the above material number 10 para there should be 7x9 not 7x8
ReplyDeleteTyping mistake friend. Corrected now.... Thanks for the update :)
Deletegood post
ReplyDeleteIBPS CWE SPL III Notification out
ReplyDeleteIBPS CWE SPL III Notiifcation came
ReplyDeletePLS EXPLAIN PARA 10
ReplyDelete