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It has been proven by F. Leong and the first author (J. Algebra (1997), 474–486) that all Moufang loops of order $p^{\alpha }{q}_1^{{\beta }_1}{q}_2^{{\beta }_2}···q_n^{{\beta }_n}$ where $p$ and $q_i$ are odd primes, are associative if $p<q_1<q_2<···<q_n$, and (i) $\alpha \le 3$ \alpha \le 3 ...