Download to find the square of any two digit number

TO FIND THE SQUARE OF ANY TWO
DIGIT NUMBER
1. The Square of a number ending in 5 may be formulated as follows:
Ex:1 Square of 25
The number next to 2 is 3, so 2 ∗ 3 = 6
The Square of 5 is 25
So, The Square of 25 is 625
Ex:2 Square of 45
The number next to 4 is 5, so 4 ∗ 5 = 20
The Square of 5 is 25
So, The Square of 45 is 2025
Hence, to find the square of any two numbers ending in 5 is to multiply the
preceding digit by the consecutive number and just put 25 to its side.
2. The Square of any number ending in 0 may be formulated as follows:
Ex:1 Square of 30
Double the 0 ---- 00
Take the number in front of 0 i.e. 3 – square it. i.e. 3 ∗ 3 = 9
Put the doubled 0 along with 9 i.e. 900
So, The Square of 30 is 900.
Ex:2 Square of 60
Double the 0 ---- 00
Take the number in front of 0 i.e. 6 – square it. i.e. 6 ∗ 6 = 36
Put the doubled 0 along with 36 i.e. 3600
So, The Square of 60 is 3600.
Hence, to find the square of any two digit numbers ending in 0 is to multiply the
number in front of 0, by itself i.e. square it and just place 00 to its side
TO FIND THE SQUARE OF ANY NUMBER
Number
Square of number
0
0
1
1
Difference
1
3
2
4
5
3
9
7
4
16
9
5
25
6
36
7
49
11
15
8
64
17
9
81
19
10
100
21
11
121
23
12
144
25
13
169
27
14
196
29
15
225
31
16
256
Hence the square of one number is different from the square of another only by
a odd number.
Square of 6 is 36
Square of 7 is 49
Hence the difference between the square is 13 which is a odd number,
Thus, odd number 13 can be derived directly by adding 5 and 7 which is two
consecutive number.
Ex:1 Square of 68
We know that square of 70 is 4900
682
137 (68 + 69)
692
139 (69 + 70)
70
2
Hence, the odd number existing between 68 and 70 square is 137 and 139
Hence, by subtracting the odd number we get the square of 68.
i.e. 4900 – (137 +139)
= 4900 – 276
=4624
Ex:2 Square of 69
We know that square of 70 is 4900
692
139 (69 + 70)
2
70
Hence, the odd number existing between 69 and 70 is 139
Hence, by subtracting the odd number we get the square of 69.
i.e. 4900 –139
= 4761
Ex:3 Square of 67
We know that square of 65 is (6 * 7 ) 25=4225
652
131 (65 + 66)
662
133 (66 + 67)
67
2
Hence, the odd number existing between 65 and 67 square is 131 and 133
Hence, by adding the odd numbers to 65 square i.e. 4225 we get the square of
67.
i.e. 4225 + (131 +133)
= 4900 + 264
=4487
Ex:4 Square of 66
We know that square of 65 is (6 * 7 ) 25=4225
652
131 (65 + 66)
672
Hence, the odd number existing between 65 and 66 is 131
Hence, by adding the odd number to 65 square
i.e. 4225+131
= 4356
The method of finding can be resolved to a formula:
Square of 27
252 = 625
= 252 + [ (2 * 25) + 1] + [ (2*25) + 3 ]
Let the nearest chosen number be N
= N2 + [ (2N) + 1] + [ (2N) + 3 ]
The rule can be extended for any number of items.
Square of 42
Chosen Number is 40(i.e. N=40) its square is 402 = 1600.
= N2 + [ (2N) + 1] + [ (2N) + 3 ]
= 402 + [ (2 * 40) + 1] + [ (2*40) + 3 ]
= 1600 + 81 +83
= 1600 +164
=1764
Square of 78
Chosen Number is 75(i.e. N=75) its square is 752 = 5625.
= N2 + [(2N) + 1] + [(2N) + 3] + [(2N) + 5]
= 752 + [(2 * 75) + 1] + [(2*75) + 3] + [(2*75) + 5]
= 5625 + [151] + [153] + [155]
= 5625 + 459
= 6084
This can be still simplified as
5625 + 3(150) + (1+3 +5)
5625 +3(150) + 32
Square of 78
Let the difference between the chosen number and given number be taken as K
Let the given number be C (i.e. 78)
C = 78
Let the chosen number be N (i.e. 75) N = 75
Let the difference between given number and chosen number be K (i.e. 3) K = 3
K=C – N
C2
=
N2
+ K * 2N +
Given number
chosen number
782
752
=
This is nothing but
K2
Difference between given & chosen
+ 3 * 2 * 75 +
a2 + 2ab + b2 = (a+b)2
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