  We bound from below the number of shifted primes p+s<x that have a divisor in
a given interval (y,z]. Kevin Ford has obtained upper bounds of the expected
order of magnitude on this quantity as well as lower bounds in a special case
of the parameters y and z. We supply here the corresponding lower bounds in a
broad range of the parameters y and z. As expected, these bounds depend heavily
on our knowledge about primes in arithmetic progressions. As an application of
these bounds, we determine the number of shifted primes that appear in a
multiplication table up to multiplicative constants.
