Problem to Brilliant ! 6

It's quite obvious that the answer will be the largest prime number less than $105000$ , because every larger number than the prime and $\leq 105000$ will have more numbers which aren't coprime.

This is proved using the fact that for primes $p_i$ , $\varphi (p) = p-1$ and if any number $x - p_1 ^{a_1} \times p_2^{a_2} \times ... p_n^{a_n}$ , then $\varphi(x) = \prod (p_i ^{a_i} -p )$

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The largest prime number $\leq 105000$ is $\boxed{104999}$ .