I began studying primes after seeing how they were used in the RSA public-key cryptography algorithm. This led me to Euler’s totient function and the idea of relative primes (on integers and on sets of integers, with their parwise coprime goodness).
And what would primes be with factoring? Ah, factorization. Why you gotta be so difficult?
But not so fast factorization, we’re building quantum computers and we’ve got Shor’s algorithm up our sleeves.
So, I think that’s the direction I’d like to trace.