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Difference between revisions of "2006 Cyprus MO/Lyceum/Problem 5"
(Standardized answer choices; minor edits)
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If both integers <math>\alpha,\beta</math> are bigger than 1 and satisfy <math>a^7=b^8</math>, then the minimum value of <math>\alpha+\beta</math> is
 
If both integers <math>\alpha,\beta</math> are bigger than 1 and satisfy <math>a^7=b^8</math>, then the minimum value of <math>\alpha+\beta</math> is
  
−
A. <math>384</math>
+
<math>\mathrm{(A)}\ 384\qquad\mathrm{(B)}\ 2\qquad\mathrm{(C)}\ 15\qquad\mathrm{(D)}\ 56\qquad\mathrm{(E)}\ 512</math>
  
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B. <math>2</math>
+
==Solution==
−
 
+
Since <math>b</math> is greater than <math>1</math> and therefore not equal to zero, we can divide both sides of the equation by <math>b^7</math> to obtain <math>a^7/b^7=b</math>, or
−
C.  <math>15</math>
+
<cmath>
−
 
+
\left(\frac{a}{b}\right)^7=b
−
D. <math>56</math>
+
</cmath>
 +
Since <math>b</math> is an integer, we must have <math>a/b</math> is an integer.  So, we can start testing out seventh powers of integers.
  
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E. <math>512</math>
+
<math>a/b=1</math> doesn't work, since <math>a</math> and <math>b</math> are defined to be greater than <math>1</math>. The next smallest thing we try is <math>a/b=2</math>.
  
−
==Solution==
+
This gives <math>b=(a/b)^7=2^7=128</math>, so <math>a=2b=2(128)=256</math>. Thus, our sum is <math>128+256=\boxed{384}</math>.
−
Since <math>b</math> is greater than <math>1</math> and therefore not equal to zero, we can divide both sides of the equation by <math>b^7</math> to obtain <math>a^7/b^7=b</math>, or
 
−
<cmath>\left( \frac{a}{b} \right) ^7=b</cmath>
 
−
Since <math>b</math> is an integer, we must have <math>a/b</math> is an integer.  So, we can start testing out seventh powers of integers. 
 
−
<math>a/b=1</math> doesn't work, since <math>a</math> and <math>b</math> are defined to be greater than <math>1</math>.  The next smallest thing we try is <math>a/b=2</math>.  This gives <math>b=(a/b)^7=2^7=128</math>, so <math>a=2b=2(128)=256</math>. Thus, our sum is <math>128+256=\boxed{384}</math>.
 
  
 
==See also==
 
==See also==

Revision as of 10:44, 27 April 2008

Problem

If both integers $\alpha,\beta$ are bigger than 1 and satisfy $a^7=b^8$, then the minimum value of $\alpha+\beta$ is

$\mathrm{(A)}\ 384\qquad\mathrm{(B)}\ 2\qquad\mathrm{(C)}\ 15\qquad\mathrm{(D)}\ 56\qquad\mathrm{(E)}\ 512$

Solution

Since $b$ is greater than $1$ and therefore not equal to zero, we can divide both sides of the equation by $b^7$ to obtain $a^7/b^7=b$, or \[\left(\frac{a}{b}\right)^7=b\] Since $b$ is an integer, we must have $a/b$ is an integer. So, we can start testing out seventh powers of integers.

$a/b=1$ doesn't work, since $a$ and $b$ are defined to be greater than $1$. The next smallest thing we try is $a/b=2$.

This gives $b=(a/b)^7=2^7=128$, so $a=2b=2(128)=256$. Thus, our sum is $128+256=\boxed{384}$.

See also

2006 Cyprus MO, Lyceum (Problems)
Preceded by
Problem 4 		Followed by
Problem 6
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
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