Moldova TST Junior 2015

[ph11]

Do there exist $ 15$ positive integers, upon increasing each of them by $ 1$,their product increases exactly $ 2015$ times?
Hint:$2015=3\cdot13\cdot31$ and $(1+1)^k=2^k$.

[ИСН]

15 is an impossibly huge number, almost beyond the grasp of human imagination. Let's consider 2. Are there two such integers that under similar treatment would give us a 3-fold increase? Huh? What about 5-fold?