primes with roots

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mberkowitz: Senior | Next Rank: 100 Posts; Posts: 76; Joined: Sun Jul 20, 2008 10:47 am; Location: new york city; Thanked: 1 times

by mberkowitz » Fri Sep 26, 2008 4:47 pm

if 2^m= 3^n, then m must equal n, and they must both equal zero.

is it the case that two different primes raised to different powers will never equal eachother?

thanks very much. i think this is a concept that everyone should know, ive personally seen multiple problems testing it.

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Azntycoon: Senior | Next Rank: 100 Posts; Posts: 31; Joined: Sun Jul 27, 2008 12:41 pm; Location: New York, NY; Thanked: 2 times

It's all about the factors

by Azntycoon » Fri Sep 26, 2008 8:23 pm

This is how I see it:

Primes, by definition, only have 2 factors (1 and itself). Therefore, 2 different prime numbers will never have the same multiples. Thus, it follows that prime numbers X and Y will never equal, even when raised by the same exponent (except for zero).

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Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 amFollowed by: 355 members; Location: Montreal; Thanked: 1090 times; GMAT Score:780

by Ian Stewart » Fri Sep 26, 2008 9:11 pm

No, that's not quite true. If you take any power of 2, say 2^3, there will be some power of 3 which is equal to 2^3, something close to 3^(1.9). What you are saying is true, however, if your exponents and bases are integers. If you have any equation of integers like the following (I'll assume n and m aren't negative though that doesn't actually matter):

2^n = 3^m

then the primes which divide the left side must also divide the right. Since 2 clearly does not divide the right side, 2 also cannot divide the left, which ensures that n=m=0.

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Azntycoon: Senior | Next Rank: 100 Posts; Posts: 31; Joined: Sun Jul 27, 2008 12:41 pm; Location: New York, NY; Thanked: 2 times

by Azntycoon » Sat Sep 27, 2008 4:01 am

Ian Stewart wrote:No, that's not quite true. If you take any power of 2, say 2^3, there will be some power of 3 which is equal to 2^3, something close to 3^(1.9). What you are saying is true, however, if your exponents and bases are integers. If you have any equation of integers like the following (I'll assume n and m aren't negative though that doesn't actually matter):

2^n = 3^m

then the primes which divide the left side must also divide the right. Since 2 clearly does not divide the right side, 2 also cannot divide the left, which ensures that n=m=0.

You're right, but I think the only conditions that need to hold true are that
a) The exponents are integers, and
b) The exponents do not equal zero.

I believe that prime bases are integers, by definition.

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