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Factor out the expression.

$2^{28}(2^{6}-1)$

$2^{28}(64-1)$

$2^{28}*63$

Since all the factors of $2^{28}$ are 2's, for now, the smallest prime factor is 2.

Next factor out 63 into $3^{2}*7$. 

The prime factors in this expression are 2,3, and 7 now.

7, being the largest, is the greatest prime factor of this expression.