Math Contest Question: Exponents

rpk5024

New member
Joined
Nov 12, 2006
Messages
2
3^8 • 5^2 = a^b , where both a and b are positive integers, find the smallest possible value for a + b.

405^2 gives 407 as an answer we figured out, but I'm not sure how to show how I got there and it seems too simple to be the best answer.


Thats the question. I was wondering how to do the question ie method and what the answer would be. Any help is appreciated.
 

stapel

Super Moderator
Staff member
Joined
Feb 4, 2004
Messages
15,943
rpk5024 said:
3^8 • 5^2 = a^b , where both a and b are positive integers, find the smallest possible value for a + b.

405^2 =407 we figured out
But 405<sup>2</sup> = (405)(405) = 164025, not anywhere close to 407 (which is just 405 plus 2, not raised to the power 2).

Do you perhaps mean "a = 405, b = 2, so a<sup>b</sup> = 405<sup>2</sup>, and a + b = 405 + 2 = 407"...?

Since the prime factor 5 occurs only twice, any power on a base containing 5 has a factor can be no greater than 2, since any larger power would return too many copies of the factor 5. So, from an algebraic standpoint, the solution "a = 405, b = 2" would seem the best possible.

Eliz.
 

rpk5024

New member
Joined
Nov 12, 2006
Messages
2
Ok thanks, any algebraic method that is purely algebraic would be nice.


ie: disproving all of the other solutions would not be correct (or at least get me a 5/5)
 
Top