= Solution
For every <prime> factor $p_i$ of $N$, <Fermat's little theorem> gives $a^{p_i-1}\equiv1\pmod{p_i}$. Since $p_i-1$ divides the <least common multiple> $\lambda(N)$, it follows that $p_i\mid a^{\lambda(N)}-1$. The <primes> are distinct and hence pairwise <coprime>, so their product divides this integer. Thus
$$
\boxed{a^{\lambda(N)}\equiv1\pmod N.}
$$
For the specified product, $N=1729$ and $\lambda(N)=\operatorname{lcm}(6,12,18)=36$. Since $N-1=1728=48\cdot36$, raising the preceding <modular congruence> to the forty-eighth power proves $a^{N-1}\equiv1\pmod N$ for every $a$ <coprime> to $N$. This is the defining phenomenon of a <Carmichael number>.
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