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Suppose t is a positive integer such that lcm[12,t]^3=(6t)^3 . What is the smallest possible value for t?

 Nov 29, 2020
 #1
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 lcm[12,t]^3=(6t)^3 

 

t = 2

 

LCM[12, 2] =12^3 =1,728

(6*2)^3 = 12^3     =1,728

 Nov 29, 2020
 #2
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Suppose t is a positive integer such that   \(lcm[12,t]^3 = (6t)^3\) . What is the smallest possible value for t?

 

suppose the lowest common multiple is 12,  (it can't be any smaller).  This would be true if t=1,2,3,4,6, or 12

 

\(12^3=(6t)^3\\ (2*6)^3=6^3t^3\\ 2^3=t^2\\ t=2\)

 

The smallest value of t is 2

 Nov 29, 2020

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