KVPY #3

Suppose that \(a_2, a_3, a_4, a_5, a_6, a_7\) are integers such that \[\frac57=\frac{a_2}{2!}+\frac{a_3}{3!}+\frac{a_4}{4!}+\frac{a_5}{5!}+\frac{a_6}{6!}+\frac{a_7}{7!},\] where \(0 \leq a_j < j\) for \(j=2, 3, 4, 5, 6, 7.\)

What is the sum \(a_2+a_3+a_4+a_5+a_6+a_7?\)

\(\)
Notation: \(!\) is the factorial notation. For example, \(8! = 1\times2\times3\times\cdots\times8 \).

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