Divisibility Of Power Differences

Number Theory Level 4

Let \( a \) and \( b \) be positive integers with \(a>b\) such that \[ 7!\Big|\big(x^a-x^b\big) \] for all integers \( x.\)

Find the smallest possible value of \( a+b.\)

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

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