Question: Prove that each positive integer n has a unique representation in the form n = a_1 1! + a_2 2! + a_3 3! +

Prove that each positive integer n has a unique representation in the form n = a_1 1! + a_2 2! + a_3 3! + · · · + a_t t! for some positive integer t and some integers a1, a2, . . . , at where at 6= 0 and 0 ≤ ai ≤ i for 1 ≤ i ≤ t. (For instance, 61 = 1! + 0 · 2! + 2 · 3! + 2 · 4!. Strong induction is useful for at least one part of this proof.

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