Let x be the first positive integer.

Then,

x = 0 (mod 2)

x+1 = 0 (mod 3)

x+2 = 0 (mod 5)

x+3 = 0 (mod 7)

x+4 = 0 (mod 11)

Hence,

x = 0 (mod 2)

x = -1 (mod 3)

x = -2 (mod 5)

x = -3 (mod 7)

x = -4 (mod 11)

Write in normal form,

x = 0 (mod 2)

x = 2 (mod 3)

x = 3 (mod 5)

x = 4 (mod 7)

x = 7 (mod 11)

And the moduli are relatively prime to each other, that is exactly the form we need.