Which is a way to use prime factorization to find the least common multiple (LCM) of 9 and 12?
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Given 
subject to the constraint 
Let
.
The gradient vectors of
and 
are:
and 
By Lagrange's theorem, there is a number
, such that
It can be seen that has local extreme values at the given region.
Let
The gradient vectors of
By Lagrange's theorem, there is a number
It can be seen that