Difference between revisions of "User talk:Bobthesmartypants/Sandbox"
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Because <math>n</math> is not divisible by <math>3</math> (therefore <math>9</math>) since <math>n>11</math> and <math>n</math> is prime, it follows that <math>n|\underbrace{111\cdots 111}_{x\text{ times}}</math>. <math>\Box</math> | Because <math>n</math> is not divisible by <math>3</math> (therefore <math>9</math>) since <math>n>11</math> and <math>n</math> is prime, it follows that <math>n|\underbrace{111\cdots 111}_{x\text{ times}}</math>. <math>\Box</math> | ||
+ | ==Picture 1== | ||
+ | <asy>draw(Circle((1,1),2)); | ||
+ | draw(Circle((sqrt(2),sqrt(3)/2),1)); | ||
+ | dot((8/5,2/5)); | ||
+ | dot((1,1)); | ||
+ | draw((1,1)--(8/5,2/5),linetype("8 8")); | ||
+ | label("a",(6/5,7/10),SSW); | ||
+ | draw((8/5,2/5)--(12/5,-2/5),linetype("8 8")); | ||
+ | label("b",(2,0),SSW); | ||
+ | </asy> | ||
+ | <cmath>\text{Find the probability that }b>a \text{.}</cmath> |
Revision as of 21:10, 19 October 2013
Bobthesmartypants's Sandbox
Solution 1
First, continue to hit
at
. Also continue
to hit
at
.
We have that . Because
, we have
.
Similarly, because , we have
.
Therefore, .
We also have that because
is a parallelogram, and
.
Therefore, . This means that
, so
.
Therefore, .
Solution 2
Note that is rational and
is not divisible by
nor
because
.
This means the decimal representation of is a repeating decimal.
Let us set as the block that repeats in the repeating decimal:
.
( written without the overline used to signify one number so won't confuse with notation for repeating decimal)
The fractional representation of this repeating decimal would be .
Taking the reciprocal of both sides you get .
Multiplying both sides by gives
.
Since we divide
on both sides of the equation to get
.
Because is not divisible by
(therefore
) since
and
is prime, it follows that
.
Picture 1