Equality modulo $p$ and general equality

Let $n\geqslant2$ be a positive integer and $p_{1},\dots,p_{r}$ its distinct prime factors.

Let $a,b$ be two positive integers such that $a\equiv b\,\,[p_{i}]$ for all $i=1,\dots,r$ and $a,b\leqslant n$ .

What necessary and sufficient condition on $n$ makes these congruences^FR equivalent to the relation $a=b$ ?

Say that $a\equiv b\,\,[p]$ for all prime $p$ with $a,b$ positive integers. Is it necessary that $a=b$ ?

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