HMMT 2016 C5

Let $a, b, c, d, e, f$ be integers selected from the set $\{1,2, \ldots, 100\}$ , uniformly and at random with replacement. Set

$M=a+2 b+4 c+8 d+16 e+32 f$

What is the expected value of the remainder when $M$ is divided by 64 ?

Dataset text NuminaMath olympiads. Quoted with source.

Show the answer

\frac{63}{2}

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