SQUARE
What is Squares?
3 x 3 = 9, 4 × 4 =16 9 and 16 are
called the square of 3 and 4.
11 × 11 = 112 = 121
121 is called the square of 11.
Perfect Square:- A natural number is called perfect square or a
square number if it is the square of a natural number.
Some examples off perfect squares are 4 = 2 × 2, 9 = 3 × 3, 25 =5 × 5 and 81 = 9 × 9
Some examples off perfect squares are 4 = 2 × 2, 9 = 3 × 3, 25 =5 × 5 and 81 = 9 × 9
The numbers 4, 9, 16, 25, 36, 49 are
called perfect squares.
Point to Remember:- All natural numbers which lie between two
perfect squares are not perfect squares. For examples, the numbers lying
between the perfect squares 36 and 49 are 37, 38, 39, 40, 41………..47, 48. All of
these numbers are not perfect squares.
Here is a table of Squares of first 20
natural numbers.
Number
|
Square
|
Number
|
Square
|
Number
|
Square
|
Number
|
Square
|
1 =
1 ×1
|
1
|
6 =
6× 6
|
36
|
11 =
11 × 11
|
110
|
16 =
16 × 16
|
256
|
2 =
2 ×2
|
4
|
7 =
7× 7
|
49
|
12 =
12 ×12
|
144
|
17 `=
17 × 17
|
289
|
3 =
3 × 3
|
9
|
8 =
8 × 8
|
64
|
13 =
13 × 13
|
169
|
18 =
18 × 18
|
324
|
4 =
4 × 4
|
16
|
9 =
9× 9
|
81
|
14 =
14 × 14
|
196
|
19 =
19 × 19
|
361
|
5 =
5 × 5
|
25
|
10 =
10 × 10
|
100
|
15 =
15 × 15
|
226
|
20 =
20 × 20
|
400
|
Interesting Patterns involving Squares
Numbers:
1:-Relationship between Triangular Numbers and
Squares Numbers:-
The sum of two consecutive triangular
numbers is a square number. Let us understand this pattern. Numbers whose dot
patterns can be arranged as triangles are known as triangular numbers. The
examples of triangular numbers are: - 1, 3, 6, 10, 15, 21
0
0 0 0
0 0 0 0 0
0
0 0
0 0 0 0 0 0
0 0
0 0
0 0 0 0
0 0 0
0 0 0
0 0 0
1 3 6 10
15
Numbers, whose dot patterns can be
arranged as squares as shown below, are known as square numbers.
0 0
0 0 0
0
0 0 0 0
0 0 0
0 0 0 0 0
0 0
0 0 0
0
0 0 0
0 0 0 0
0 0 0
0 0 0
0
0 0 0
0 0 0
0 0
0 0
0 0 0
0
1 +3 = 22 3 +
6 = 32 6 + 10 = 42
10 + 15 = 25 = 52
As it is clear from the above dot
patterns, we get a square number by adding two consecutive triangular numbers.
Numbers between Consecutive Square Numbers:-
This pattern of square numbers helps
you quickly find out the number of non-square numbers between two consecutive
square numbers. Let us observe some examples of non-square numbers between two
consecutive square numbers in the following tables.
Squares of
Natural Consecutive Numbers
|
Nonsquares
between Consecutive Square Number
|
Numbers of
Nonsquare Numbers between Consecutive Square Numbers
|
1 and 2; 12 = 1, 22 =
4
|
Between 1 and 4:- 2, 3
|
2 = 2x 1
|
2 and 3; 22 = 4, 32 =9
|
Between4 and 9; 5,6,7,8
|
4 = 2 x2
|
3 and 4; 32 =9, 42 =
16
|
Between 9 and 16; 10, 11, 12, 13, 14, 15
|
6 = 2x 3
|
4 and 5; 42 = 16, 52 =
25
|
Between 16 and 25; 17, 18, 19, 20, 21, 22,
23, 24
|
8 = 2 x 4
|
n2, (n + 1)2
|
2n = 2 x n
|
From the above table, it is clear that
the number of non square numbers between the squares of two consecutive natural
numbers n and (n + 1) is equal to 2n.
This relationship is also linked with
the difference in the consecutive square numbers. We can express the difference
between two consecutive square numbers n2 and (n + 1)2 as
{(n + 1)2 – n2 }
= (n + 1 + n) (n + 1 – n) = 2n +1
Subtracting 1 from both side, we get,
{(n + 1)2 – n2 } – 1 = 2n
Therefore, the number of non square
numbers (i.e., 2n) between two consecutive square numbers n2 and (n
+ 1 )2 is 1 less than their difference.
Product of two consecutive even or odd
natural numbers:-
This pattern relates to applying the
identity a2 – b2 = (a + b)(a – b) for finding the product
of two consecutive even or odd numbers. As the difference between two
consecutive even or odd numbers is two, they can be written in the following
way for dividing out their product easily.
Two consecutive odd numbers
15 x 17 = (16 + 1)(16 – 1) = 162
-12 = 256 – 1 = 255
Two consecutive even numbers
24 x 26 = (25 + 1) (25 – 1) = 252
– 12 = 625 – 1 = 624
In general, for any odd or even
natural number n, we have
(n + 1)(n – 1) = n2 –
1
Squares of Numbers having 5 in the
Unit Place:
This pattern makes it easy to find the
squares of numbers with digit 5 in the units place. To use this pattern, you
need to write such numbers in the following way:-
152 = 225 = (1 x 2)
hundreds/ (100) + 25
252 = 625 = (2 x 3)
hundreds/ (100) + 25
352 = 1225 = (3 x 4)
hundreds/ (100) + 25
452 = 2025 = ( 4 x 5)
hundreds/(100) + 25
552 = 3025 = (5 x 6)
hundreds/ (100) + 25
And so on. you can get square of any
number having 5 at unit place.
In general, any number, say a5 whose unit’s
digit is 5, can be written as
a5 = 10a + 5
(since a is at the tens place)
Squaring the given number, we get
(a5) 2 = (10a + 5)2
= (10a + 5) (10a +5)
= 10a (10a + 5) (10a + 5)
= 10a2 + 50a + 50a + 25
= 100a2 + 100a + 25
= 100a (a + 1) + 25
= a (a + 1) hundreds + 25
You have got the general form of the
pattern for finding the square of any number having 5 at the unit place.
Using the above pattern, we can find
out the square of 85 I the following way:-
852 = (8 x 9) 100 + 25 =
7225
Some more pattern showing interesting
number sequences, are given following.
When 7 are at unit place:-
(7)2 = 49,
(67)2 = 4489,
(667)2 = 444889
(6667)2 = 44448889
(66667)2 = 4444488889
and so on.
When 1 is at unit place in special cases:-
(11)2 = 121
(101)2 = 10201
(10101)2 = 102030201
(1010101)2 = 1020304030201
and so on.
PROPERTY OF PREFECT SQUARES:- There are some useful and interesting property
of perfect squares.
1. The square of an even number is
even.
2. The square of an odd number is odd.
These two properties follows from the
general rule of multiplication that the product of two odd numbers is always an
odd number and the product of two even numbers is always even.
3. Perfect squares end in even number
of zeroes. Conversely, a number ending in odd number of zeroes cannot be a
perfect square.
For example:- The numbers 100, 400 and
1,60,000 are perfect squares.
The number 1000 and 4, 00,000 are not
perfect squares.
Note:-
This property can only be used to conclude that a given number is not a perfect
square. It does not mean that all numbers which end in even number of zeroes
are perfect squares. For example, numbers 200 and 30,000 are ending in even
number of zeroes, but they are not perfect squares.
4. A number ending with digits 2,3, 7
or 8 is not a square.
For example:- the number 122, 153,
4267 and 1568 are not perfect squares as they have the digits 2,3,7 or 8 in the
units place.
Note: Property 4 can be used only to
conclude that a given number is not a perfect square. It does not imply that
numbers which do not end in digits 2, 3, 7, 8 are perfect squares. For example,
the numbers 59, 104, 86, and 91 are not ending in digits 2 or 3 or7 or 8 but
none of them is a perfect square.
5. A perfect square leaves a remainder
of 1 or 0 when divided by 3; e.g. if 14 is divided by 3, the remainder is 0.
If 256 is divided by 3, the remainder
is 1.
Note:
This property is more useful to conclude that a given number is not a perfect
square than to prove that it is a perfect square.
6. If n is a perfect square, then 2n
can never be a perfect square as 2 is not a perfect square. However, 4n is a
perfect square since 4 is a perfect square. Hence, we can conclude that the
product of two perfect squares is a perfect square.
7. Pythagorean Triplets: - Three natural numbers a, b, c are said to
form a Pythagorean triplet if a2 + b2 = c2.
Also for every natural number m > 1, (2m, m2 – 1, m2 +
1) form a Pythagorean triplet.
8. This property helps you easily find
the difference between two consecutive squares. If you have two consecutive
natural numbers n and (n + 1), then the difference between their squares is
given by
(n + 1)2 – n2 =
n + 1 + n
This property is obtained by using the
identity
a2 – b2 = (a + b)
(a – b)
(n + 1)2 – n2 =
(n + 1 + n)(n + 1 – n)
= (n + 1 + n)
Example:- (38)2 – (37)2
= (38 + 37) = 75
Actually, finding the squares of 38
and 37, we get
382 = 1444
372 = 1369
Difference is 1444 – 1369 = 75
9. The sum of the first n odd natural
numbers is equal to n2
1 = 12
1 + 3 = 22
1 + 3 +5 = 32
1 + 3 + 5 +……..+ (2n – 1) = n2
Finding Whether a Given Number is a
Perfect Square:-
Look for the following properties in a
given number to quickly dete4rmine whether the number is perfect square.
1. If the given number ends in odd
number of zeroes, it is not a perfect square.
2. If the given number has the digit
2, 3, 7 or 8 in the units place, it is not a perfect square. 3. If the given
number does not leave a remainder of 0 or 1 when divided by 3, it is a perfect
square.
3. If the given number does not leave
a remainder of 0 or 1 2h3n divided by 3, it is not a perfect square.
4. If the given number can be
expressed as a product of pairs of all its factors, it is a perfect square.
In the following examples, 144 is a
perfect square and 500 is not a perfect square because 144 can be expressed as
a product of pairs of all its factors.
a. 144 = 2 x2 x 2 x 2 x 3 x 3 = ( 2 x
2 x 3) x (2 x 2 x 3)
144 is expressed as the product of
pairs of all its factors, hence, 144 is perfect square.
b. 500 = 2 x 2 x 5 x 5 x 5
Making pairs of the factors of 500, we
find that factor 5 does not have a pair, hence 500 is not a perfect square.
Example 1:- Find the smallest number by which 1008
should be multiplied to make it a perfect square.
Solution:- Find the factors of 1008.
2|1008|
2|504|
2|252|
2|126|
3|63|
3|21|
7|7|
_|1|
or 1008 = 2 x 2 x 2 x 2 x 3 x 3 x 7
= 2 x 2 x 2 x 2 x
3 x 3 x 7
Making pairs of the factors of 1008,
we find that 7 does not have a pair.
If 1008 is multiplied by 7, then all
the factors of 1008 will become pairs and the number so obtained will be a
perfect square.
Therefore, the smallest number by
which 1008 should be multiplied to make it a perfect square is 7.
Example2:- find the smallest number by which 512
should be divided to make it a perfect square.
Solution:- Factoring 512, we get,
2|512|
2|256|
2|128|
2|64|
2|32|
2|16|
2|8|
2|4|
2|2|
_|1|
512 =
2 x 2x 2 x 2 x 2 x 2 x 2 x 2 x 2
Making pairs, we find that the last 2
does not have a pair. If 512 is divided by 2, all the factors will have pairs
and the number will be a perfect square.
Example3:- Find the least square number which is
divisible by 6,9, and 15.
Solution:- The L.C.M.
OF 6, 9 and 15 is 90
90 is the least number divisible by 6,
9 and 15. By prime factorization, we get
90 = 2 x 3 x 3 x 5
To make it a perfect square, it must
be multiplied by (2 x 5) = 10
i.e., 90 x10 = 900
Therefore, 900 is the smallest perfect
square divisible by 6,9 and 15.
Example 4:- Find
the Pythagorean triplets whose smallest member is 18
Solution: - If 2m, m2 – 1, m2 +1 form a Pythagorean triplet,
then 2m is the smallest member of this triplet.
So, 2m = 18
m = 9
Therefore, m2 – 1 = 81 – 1 = 80
And m2 + 1 = 81 + 1 = 82
Hence, the Pythagorean triplets are
18, 80 and 82.
Example 5 :- Find the smallest number by which 2904 must be
divided to make it a perfect square.
Solution:- Factorizing 2904, we get
2|2904|
2|1452|
2|726|
3|363|
11|121|
11|11|
__|1|
2904 = 2 x 2 x 2 x 3 x 11 x11
Making pairs of the factors we find
that 2 and 3 are not paired. Therefore, the smallest number by which 2904 must
be divided to make it a perfect square is
2 x 3 = 6
2904 /6 =484
484 = 2 x 2 x 11 x 11,
Which is a perfect square?
Example 6:- Is 6615 a perfect square? If not, find the
smallest number by which 6615 has to be multiplied to make it a perfect square.
Solution:- Factorizing 6615, we get
6615 = 3 x 3 x 3 x 5 x 7 x 7
Since all the factors of 6615 are not
in pairs, therefore, 6615 is not a perfect square.
3 and 5 are the factors which are not
in pairs. Therefore, multiplying 6615 with 3 x 5 will complete the pair and the
product obtained will be a perfect square.
6615 x 3 x 5 = 99225
99225 = 3 x 3 x 3 x 3 x 5 x 5 x 7 x 7
Since, all factors of 99225 occur in
pairs, it is a perfect square.
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