Question

If$a,b$, are integers such thatlocalid="1646129199316" $a\equiv b\left(modp\right)$for every positive prime$p$, prove that $a=b$.

Short answer

It is proved that if $a$and$b$ are integers such that$a=b\left(modp\right)$ for every positive prime$p$ then $a=b$.

Step-by-step solution

Consider the given situation

Let us consider the given situation$a=b\left(modp\right)$ for every prime.

Assume$a\ne b$ , this implies that $a-b\ne 0$. Therefore,$a-b$ is some finite number.

Prove that a=b

Let us consider finite prime factors of$a-b$ as$p_1,p_2,.....,p_k$ .

As prime numbers are infinite, consider $p_{k+1}$be other prime from$p_1,p_2,......,p_k$ and$p_{k+1}$ is not a prime factor of$a-b$ .

This implies that $p_{k+1}/\left(a-b\right)$contradictsthe assumed condition,that is $a=b\left(modp\right)$, for every prime.

This implies that our assumption is wrong that $a\ne b$. Hence $a=b$.

Thus, it is proved that if $a$and $b$are integers such that$a=b\left(modp\right)$ for every positive prime $p$then$a=b$ .