Maths Olympiad Prep

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Number theory Difficulty 5.3 AIME, harder Find the answer

14. A Cantor expansion of a positive integer nn is a sum
n=amm!+am−1(m−1)!+⋯+a22!+a11!n=a_{m} m!+a_{m-1}(m-1)!+\cdots+a_{2} 2!+a_{1} 1!
where each aja_{j} is an integer with 0⩽aj⩽j0 \leqslant a_{j} \leqslant j.
a) Find Cantor expansions of 14,56 , and 384 .
b) Show that every positive integer has a unique Cantor expansion.

A number or a short expression. Spacing and $ signs are ignored.

Solutions — 2

Solution 1

14. а) 14=2⋅3!+1⋅2!,56=2⋅4!+1⋅3!+1⋅2!,384=3⋅5!+1⋅414=2 \cdot 3!+1 \cdot 2!, 56=2 \cdot 4!+1 \cdot 3!+1 \cdot 2!, 384=3 \cdot 5!+1 \cdot 4 !

Solution 2

14. a) 14=2⋅3!+1⋅2!,56=2⋅4!+1⋅3!+1⋅2!,384=3⋅5!+1⋅4!14=2 \cdot 3!+1 \cdot 2!, 56=2 \cdot 4!+1 \cdot 3!+1 \cdot 2!, 384=3 \cdot 5!+1 \cdot 4!

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.