INMO - 2012

Outline:

Note that we must have the numbers be $p, p+2, p+6, p+8$ because if there was three consecutive twin primes then one of them would have to be a multiple of $3$ . This also means $p+4\equiv 0\pmod{3}$ so $p\equiv 2\pmod{3}$ .

If we subtract $p_1$ and $q_1$ , then, the result would equal $2-2=0\pmod{3}$ .

Similarly, we can find that $p+4\equiv 0\pmod{5}$ or else one of the primes must be divisible by $5$ . So $p\equiv 1\pmod{5}$ , and subtracting $p_1$ and $q_1$ , we see that the result is $1-1=0\pmod{5}$ .

also trivially, $p_1-q_1\equiv 0\pmod{2}$ .

Thus, $p_1-q_1\equiv 0\pmod{30}$ and the desired answer is $\boxed{30}$