# 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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