Digit Nine Always Remain

For Any Number of Digits, Digit Nine Always Remain
For Two Numbers (10 a+b) and (10 x+y)
For two digits numbers, numbers can be written in four ways like
this
1. (10 a+b) and (10 x+y)
2. (10 a+b) and (10 y+x)
3. (10 b+a) and (10 x+y)
4. (10 b+a) and (10 y+x)
Multiple with each other’s
 (10 a+b)*(10 x+y) = 100 ax + 10 bx + 10 ay +by
 (10 a+b)*(10 y+x) = 100 ay + 10 by + 10 ax + x b
 (10 b+a)*(10 x+y) = 100 bx + 10 ax + 10 by + a y
 (10 b+a)*(10 y+x) = 100 by + 10 ay + 10 bx + ax
Subtract each one respectively
100 ax +10 bx + 10 a y + by – 100 a y – 10 by- 10 ax – x b = 90 ax + 9 bx –
90 a y – 9 by = 9(10 ax+bx-10 a y- by)
100 ax +10 bx + 10 ay + by – 100 bx – 10 ax- 10 by – a y = 90 ax – 90 bx
-9 a y – 9 by = 9(10 ax-10 bx- a y-by)
100 ax +10 bx + 10 a y + by – 100 by – 10 a y- 10 bx – ax = 99 ax – 99 by
= 9(11 ax -11 by)
100 a y + 10 by + 10 ax + x b -100 bx – 10 ax -10 by –a y = 99 ay -99bx =
9(11 a y-11 b x)
100 a y + 10 by + 10 a x + x b -100 by – 10 a y -10 b x –ax = 90 a y -90 by+
9 ax -9 x b = 9(10 a y-10 by +ax –x b)
For example
25 & 32, we can write them in four ways like 25, 32, 52 and 23
and now multiple with each other’s like
(25*32),(25*23),(52*32) and (52*23)
25*32=800
25*23=575
52*32=1664
52*23=1196
1664 – 1196 = 468 =4+6+8 =18 =1+8=9
1664 – 800 = 864 =8+6+4 =18 =1+8=9
1664 – 575 = 1089=1+0+8+9=18 =1+8=9
1196 – 800 = 396 =3+9+6 =18 =1+8=9
1196 – 575 = 621 =6+2+1=9
800 – 575 = 225 =2+2+5=9
Copyrighted to Piyush Goel