- 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 = 1002, 2500 = 502, 490000=7002
- Squares of even numbers are always even and squares of odd numbers are always odd.
- For Example : 22 = 4, 82=64, 402=1600, 52=25, 92=81, 172=289
- For any two consecutive natural numbers n and (n+1), we have
- (n+1) 2-n2 = (n+1+n)(n+1-n) = (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 X2 + y2 = z2
- for example, (6, 8, 10) is a Pythagorean triplet.
- Since 62 + 82 = 36+64 = 100 and 102 = 100
- Note : For any natural number n greater than 1, the Pythagorean triplet is given by (2n, n2-1, n2+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, m2-1, m2+1
- let m = 6
- So, 2m = 12
- m2-1 = 62-1 = 36-1 = 35
- m2+1 = 62+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 12 =1 = sum of the first 1 odd number
- 22 = 4 = 1+3 = sum of the first 2 odd numbers
-
32 =9 = 1+3+5 = sum of the first 3 odd numbers
-
52 =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,
- 52 = 25 = 12+13
- 112 = 121 = 60+61
- 412 = 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 (n-1) = n2-1
- Using this property, we can find the product of two consecutive even or odd natural numbers easily.
- 7 x 9 = (8-1) x (8+1) = 63 = 82-1
- 15 x 17 = (16-1) x (16+1) = 255 =162-1
- 24 x 26 = (25-1) x (25+1) = 624 = 252-1
- Observe the following :
- 12 = 1 and 22=4, Between 12 = 1 and 22=4, the numbers are 2 3.i.e., there are 2 x 1 = 2 non square numbers.
- 22 = 4 and 32 = 9, Between 22 = 4 and 32 = 9, the numbers are 5, 6, 7, 8 i.e., there are 2 x 2 = 4 non square numbers.
- 32 = 9 and 42 = 16, Between 32 = 9 and 42 = 16, the numbers are 10, 11, 12, 13, 14, 15. i.e., there are 2 x 3 = 6 non square numbers.
- 82 = 64 and 92=81, Between 82 = 64 and 92=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 :
- 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 = 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
