Problem:

Ancien Indian Problem: If eggs are removed form a basket 2, 3, 4, 5, and 6 at a time, there remain, respectively, 1, 2, 3, 4, and 5 eggs. But if eggs are removed 7 at a time, no eggs remain. What is the least number of eggs that could have been in the basket?

So far this is what I believe this exercise implies but I am not sure:

x ≡ 1 (mod 2)

x ≡ 1 (mod 3)

x ≡ 1 (mod 4)

x ≡ 1 (mod 5)

x ≡ 1 (mod 6)

x ≡ 0 (mod 7)

Can anyone help? Thank you!