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- Jan 27th 2011, 06:38 PM #1

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- Jan 27th 2011, 06:59 PM #2

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- Jan 27th 2011, 07:09 PM #3

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- Jan 27th 2011, 07:23 PM #4

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- Jan 27th 2011, 08:50 PM #5

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- Jan 27th 2011, 09:03 PM #6

- Jan 28th 2011, 07:53 AM #7

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Let be the greatest common divisor of and . So, is the**smallest positive integer**that can be expressed a linear combination of and .(Theorem.)

Consider the following linear combinations : , where are integers.

From the equation we can see that is the smallest positive linear combination of and precisely when is the smallest positive linear combination of and . Applying the theorem we have .