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If a, b, c and d are positive integers such that $${{\mathtt{a}}}^{{\mathtt{b}}}{\mathtt{\,\times\,}}{{\mathtt{c}}}^{{\mathtt{d}}} = {{\mathtt{2}}}^{{\mathtt{10}}}{\mathtt{\,\times\,}}{{\mathtt{7}}}^{{\mathtt{9}}}$$, what is the least possible value of a + b + c + d ?

 May 3, 2015

Best Answer 

 #3
avatar+33615 
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210 is 21*29 so I just combined the 29 with the 79 to get 149

 

I couldn't see any other combination giving a lower result (the original expression results in a sum of 28), though I guess I haven't actually proved that 26 is the smallest possible value.

.

 May 4, 2015

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 #1
avatar+33615 
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a=2, b=1, c = 14, d = 9

 

210*79 = 21*149

 

a + b + c + d = 26

.

 May 4, 2015
 #2
avatar+118608 
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How did you do this Alan?

 May 4, 2015
 #3
avatar+33615 
+10
Best Answer

210 is 21*29 so I just combined the 29 with the 79 to get 149

 

I couldn't see any other combination giving a lower result (the original expression results in a sum of 28), though I guess I haven't actually proved that 26 is the smallest possible value.

.

Alan May 4, 2015

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