The number can be written as , where a,b, and c are positive integers. Find a*b*c.

Any hints?

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- March 12th 2006, 12:38 PMJamesonAIME I Question
The number can be written as , where a,b, and c are positive integers. Find a*b*c.

Any hints? - March 12th 2006, 01:12 PMThePerfectHackerQuote:

Originally Posted by**Jameson**

Open parantheses,

=

Thus, we have,

(1)

(2)

(3)

(4)

Now multiply (2)(3)(4) to get,

Thus,

- March 12th 2006, 02:06 PMThePerfectHacker
The reason being that,

and all thus, they are not squares. Then, .

Here is an outline of the proof, to keep thing simple let us do it with only two terms,

and are positive integers with a non-square.

Now if then,

Thus,

Impossible because LHS is irrational.

If you have, you get the same thing.

Thus, which implies that

-------------- - March 12th 2006, 02:14 PMJameson
Thanks for the help :D