Many of the beginner explanations use very simple examples:
[like this](http://pajhome.org.uk/crypt/rsa/rsa.html)

It is still hard to factorize a small number by hand, compared to other operations. There is definitely a size of prime numbers where hand computation is still feasible, but hand factorizing isn't. Would you, for instance, even armed with a [list of primes](http://primes.utm.edu/lists/small/1000.txt), be able to quickly tell me what the prime decomposition of 1043 is?
>! It's 149 and 7. Slow to decompose, but easy to multiple by hand. You can go a lot bigger by hand or with an abacus etc.