For each integer $p>1$ coprime to $10$ there is a positive divisibility multiplier $m<p$ which preserves divisibility by $p$ for the following function on any positive integer, $n$:

$f(n)=(\text{all but the last digit of }n)+(\text{the last digit of }n)\cdot m$.

That is, if $m$ is the divisibility multiplier for $p$, then $f(n)$ is divisible by $p$ if and only if $n$ is divisible by $p$.

(When $n$ is much larger than $p$, $f(n)$ will be less than $n$ and repeated application of $f$ provides a multiplicative divisibility test for $p$.)

For example, the divisibility multiplier for $113$ is $34$.

$f(76275)=7627+5\cdot 34=7797$: $76275$ and $7797$ are both divisible by $113$.
$f(12345)=1234+5\cdot 34=1404$: $12345$ and $1404$ are both not divisible by $113$.

The sum of the divisibility multipliers for the primes that are coprime to $10$ and less than $1000$ is $39517$. What is the sum of the divisibility multipliers for the primes that are coprime to $10$ and less than ${10}^7$?