The LCM of 2, 3, 4, 5, 6 is 60, so 61 will give a remainder of 1 when divided by each of them. In fact, any multiple of 60 plus 1 will produce the same remainder. We can write this as 60n+1 where n is an integer.
But 60n+1 must be divisible by 7, so 60n+1=7m where m is an integer.
We can write this as (8×7+4)n+1=7m. So if 4n+1 is divisible by 7 then we can find out how many eggs.
When n=5, 4n+1=21 which is divisible by 7, so 60n+1=301=43×7 (43 packages). So the smallest number of eggs in the shipment is 301.