Number theory / Group theory: consecutive integers divisible by at least n prime numbers Claim:
There exist 15,251 successive positive integers $a_1, a_2\dots,a_{15251}$  such that
each $a_i$ where ($1\le i\le 15251$) is divisible by at least 251 different prime numbers
Is there a neat way to prove the above claim? All help is greatly appreciated.

Use the Chinese Remainder Theorem.         A good system of congruences is
$x\equiv 0 \pmod{m_0}$
$x\equiv -1\pmod{m_1}$
$x\equiv -2\pmod{m_2}$
and so on up to
$x\equiv -15250\pmod{m_{15250}}$
for pairwise relatively prime $m_i$ chosen so that the divisibility by lots of primes is satisfied.
