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Example: Interesting Nine

(related to Part: Oddities and Curiosities)

For any Number Digits here is for two Digit Number

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

• \((10 a+b)\cdot(10 x+y) = 100 ax + 10 bx + 10 ay +by\)
• \((10 a+b)\cdot(10 y+x) = 100 ay + 10 by + 10 ax + x b\)
• \((10 b+a)\cdot(10 x+y) = 100 bx + 10 ax + 10 by + a y\)
• \((10 b+a)\cdot(10 y+x) = 100 by + 10 ay + 10 bx + ax\)

Subtract each one respectively

1. \(100 ax +10 bx + 10 a y + by - 100 a y - 10 by- 10 ax - x b = 90 ax + 9 bx - 90 a y – 9 by = 9(10 ax+bx-10 a y- by)\)
2. \(100 ax +10 bx + 10 ay + by – 100 bx – 10 ax- 10 by - a y = 90 ax - 90 bx - 9 a y – 9 by = 9(10 ax-10 bx- a y-by)\)
3. \(100 ax +10 bx + 10 a y + by – 100 by – 10 a y- 10 bx - ax = 99 ax – 99 by = 9(11 ax -11 by)\)
4. \(100 a y + 10 by + 10 ax + x b -100 bx - 10 ax -10 by –a y = 99 ay -99bx = 9(11 a y-11 b x)\)
5. \(100 a y + 10 by + 10 a x + x b -100 by - 10 a y -10 b x –ax = 90 a y -90 by+ 9 ax -9 x b = 9(10 a y-10 by +ax –x b)\)

For example $25$ & $32,$ we can write them in four ways like $25, 32, 52$ and $23$ and now multiple with each other like this

\((25\cdot 32)\),\((25\cdot 23)\),\((52\cdot 32)\) and \((52\cdot 23)\) \(25\cdot 32=800\) \(25\cdot 23=575\) \(52\cdot 32=1664\) \(52\cdot 23=1196\) \(1664 – 1196 = 468 =4+6+8 =18 =1+8=9\) \(1664 – 800 = 864 =8+6+4 =18 =1+8=9\) \(1664 – 575 = 1089=1+0+8+9=18 =1+8=9\) \(1196 – 800 = 396 =3+9+6 =18 =1+8=9\) \(1196 – 575 = 621 =6+2+1=9\) \(800 – 575 = 225 =2+2+5=9\)

Nine always Remain

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