saw this following post by meiju on fb (pardon the blatant copying of text.. haha..):
What is the smallest positive prime factor of the integer: 2005^2007 + 2007^2005? - Singapore Maths Olympiad question for high school students. (the answer is 2, btw)
i couldn't help but try to prove that e answer was 2.. guess i'm still quite attracted to SMO questions.. anyway tt aside, e question does look quite intimidating.. but SMO questions, given their nature, are most of e time relatively simple, tt is if u noe wat concepts to apply.. in proving/solving this question, u need to make use of 3 simple concepts..
1. odd number * odd number = odd number
when u realise tt this is e case, u would see tt 2005^2007 is just multiplying a chain of odd numbers.. at e add of all e calculation, u would end up with an odd number.. same goes for 2007^2005..
2. odd number + odd number = even number
so e result of 2005^2007 (which is an odd number) + e result of 2007^2005 (also an odd number) gives u an even number..
3. even numbers are divisible by 2.
so ultimately, 2005^2007 + 2007^2005 would give u an even number and 2 would be one of it's factors.. coincidentally, 2 is e smallest positive prime factor possible for any given number..
without doing any real calculations, e question has been solved..
Saturday, November 28, 2009
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