# SQUARES

## SQUARES

When a number is multiplied by the same number, we get its square. For example, when 5 is multiplied by 5, we get the square of 5 and is 25.

First, let us observe certain interesting facts of squares.

8×9+9 =92 (9 square), 14×15+15 = 152 6×7+7=72 10×11+11 =112
8×9-8=82 14×15-14 =142 6×7-6=62 10×11-10=102
OR,
Any number a x (a+1) + (a+1) = (a+1)2
a x (a+1) – a = a2

8 x10 + 1 = 81 = 92 7 x 9 + 1 = 64 = 82 5 x 7 + 1 = 36 = 62
(A-1) x (A +1) +1 = a2
82 + 8 + 9 = 81 = 92 122 + 12 + 13 = 169 = 132 252 +25 +26 = 676 = 262
82 – 8 -7 = 49 = 72 122 +12 – 11 = 121 = 112 252 -25 -24 = 576 =242
A2 + A + (A+1) = (A+1) 2
A2 – A – (A-1) = (A-1) 2

To find the square of any number beginning or ending with 5:
Square the tens digit number and add to it the number other than 5. Square the unit digit number.
582 = (52+8) (82), 592= (52+9) (92), 752= (72+7) (52), 652 = (62+6)(52)
=3364 =3481 =5625 =4225

It is advisable to know squares of all numbers from 1 to 25. It will help us in finding squares of almost all numbers.

We know that (a + b) x (a – b) = a2 – b2 —-→ a2 = (a + b) x (a – b) + b2

Let us find the squares of numbers, which are near to 100:
982 = (98 + 2) x (98 – 2) + 22 = 100 X 96 + 22 = 9604
932 = (93+7) x (93-7) + 72 = 8649
872 = (87+13) x (87-13) + 132 = 7400 + 169 = 7569
1032 = (103-3) x (103+3) + 32 = 10600+9 = 10609
1152 = (115-15) x (115+15) + 152 = 13000 + 225 = 13225

Let us see the numbers, which are near 50
372 = (37+13) x (37-13) +132 = 50 x 24 +169 =1369
482 = (48+2) x (48-2) +22 = 2304

When numbers are near 50, first two digits can be found out by deducting 25 from that number. In the above example 23 = 48 – 25

Square of 25 is 625.
Square of any number ending with 25 follows following rule.
Square the number before 25, add half of it and multiply by 10. Then 625 follow that number.
Eg: 34252 is (342 +34/2)*10 followed by 625
(1156+ 34/2)*10 = 11730
34252 = 11730625
39252 = (392 +39/2)*10, followed by 625
(1521+19.5)*10 =15405 followed by 625
=15405625

A*(A+1) + (A+1) = (A +1) 2 -> 3*4+4 = 42 = 16
A* (A-1)-A = A2 -> 3*4-3 = 32 = 9
A2 + A + (A+1) = (A+1) 2 -> 52 + 5 + 6 = 62 = 36
A2 – A – (A+1) = (A-1) 2 -> 52 – 5 -4 = 42 = 16
(A+1)*(A-1) + 1 = A2 -> 5*7 + 1 = 62 = 36

A 2 = (A+B) * (A-B) + B2 -> 622 = (62 + 12) *(62 – 12) +122
=74*50 +144 =3844
972 = (97+3)*(97-3) + 32
= 100*94 +9 = 9409

For numbers near 50
482 = (48-25)(22) = 2304
622 = (62-25)(122) = 37(144) =3844

(50 + A) 2= (25+A)*100 + A2 -> 582 = (25+8)*100 +82
= 3364
(10A +5) 2 = (A)*(A+1)*100 + 25 -> 652 = 6*7*100+25
= 4225
50^2+50+51= 2601 = 51^2
18^2+18+19 = 324 +37 = 361 = 19^2
A^2+A+(A+1) = (A+1)^2
A^2-a-(a-1) = (a-1)^2
16^2+16-15 =256-31 = 225 =15^2
10.08.2016
If a number ends with 2 or 3 or 7 or 8, then it is not a perfect square number.
If the sum of all digits (digital root) to a single digit is not equal to 1.4,7 or 9, then it is not a perfect square number.