Question 300230
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Start with any number (but NOT 1)

Say {{{73}}}

Square that number:

{{{73^2 = 73*73 = 5329}}}

Subtract {{{1}}}

{{{5329 - 1 = 5328}}} 

Divide by one less than your original number.

The original number was {{{73}}}.  One less than {{{73}}} is {{{72}}}.
So dividing {{{5328}}} by {{{72}}}

{{{5328}}}{{{"÷"}}}{{{72}}}{{{"="}}}{{{74}}}

Now subtract your original number.

{{{74-73 = 1}}}

You reached {{{1}}}.

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That works for all numbers. Why does it work?  

Start with any number

Say {{{x}}}

Square that number:

{{{x*x=x^2}}}

Subtract {{{1}}}

{{{x^2 - 1}}} 

Divide by one less than your original number.

The original number was {{{x}}}.  One less than {{{x}}} is {{{x-1}}}.
So dividing {{{x^2-1}}} by {{{x-1}}}

{{{x^2-1}}}{{{"÷"}}}{{{x-1}}}{{{"="}}}{{{(x^2-1)/(x-1)=((x-1)(x+1))/(x-1)=((cross(x-1))(x+1))/(cross(x-1))=(x+1)}}}

Now subtract your original number, which was {{{x}}}

{{{(x+1)-x = x+1-x = 1}}}

You reached {{{1}}}.

Notice that you could not have started with 1 because you
cannot divide by 1 less than 1 because that would be dividing
by zero!

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You can make up a rational expression with a factor in the bottom
that will cancel a factor in the top.  Say you start with this

{{{((x-2)(x+3))/(x+3)}}}

Then multiply the top out:

{{{(x^2+x-6)/(x+3)}}}

Then to make the numerator {{{x^2+x-6}}}, you say

Start with any number (other than 2).

Square it (that gives {{{x^2}}}

Add the original number (that gives {{{x^2+x}}}

Subtract 6  (that gives {{{x^2+x-6}}}

Divide by 2 less than the original number (that gives {{{(x^2+x-6)/(x-2)=((x-2)(x+3))/(x+3)=((cross(x-2))(x+3))/(cross(x-2))=(x+3)}}}

Subtract your original number  (that gives {{{(x+3)-x=x+3-x=3}}}

You reach 3.

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Try it with an arbitrary number.

Start with any number (other than 2).  Say we start with 83

Square it (that gives {{{83^2=83*83=6889}}}

Add the original number (that gives {{{6889+83=6972}}}

Subtract 6  (that gives {{{6972-6=6966}}}

Divide by 2 less than the original number (2 less than 83 is 81.
{{{6966}}}{{{"÷"}}}{{{81}}}{{{"="}}}{{{86}}}

Subtract your original number  {{{86-83=3}}}

You reach 3.
 
You cannot start with the number 3 because you would have to
divide by 0.
 
Edwin</pre>