14. N1 (JAP) Determine all positive integers $n \geq 2$ that satisfy the following condition: For all integers $a, b$ relatively prime to $n$,  
$$ a \equiv b(\bmod n) \quad \text { if and only if } \quad a b \equiv 1(\bmod n) $$
14. The given condition is obviously equivalent to $a^{2} \equiv 1(\bmod n)$ for all integers $a$ coprime to $n$. Let $n=p_{1}^{\alpha_{1}} p_{2}^{\alpha_{2}} \cdots p_{k}^{\alpha_{k}}$ be the factorization of $n$ onto primes. Since by the Chinese remainder theorem the numbers coprime to $n$ can give any remainder modulo $p_{i}^{\alpha_{i}}$ except 0, our condition is equivalent to $a^{2} \equiv 1\left(\bmod p_{i}^{\alpha_{i}}\right)$ for all $i$ and integers $a$ coprime to $p_{i}$. Now if $p_{i} \geq 3$, we have $2^{2} \equiv 1\left(\bmod p_{i}^{\alpha_{i}}\right)$, so $p_{i}=3$ and $\alpha_{i}=2$. If $p_{j}=2$, then $3^{2} \equiv 1\left(\bmod 2^{\alpha_{j}}\right)$ implies $\alpha_{j} \leq 3$. Hence $n$ is a divisor of $2^{3} \cdot 3=24$. Conversely, each $n \mid 24$ has the desired property.