When Googol (10¹⁰⁰) is divided by 1,2,3,4,5,6... the remainders form the sequence 0,0,1,0,0,4...

When Googolplex (10¹⁰¹⁰⁰) is divided by 1,2,3,4,5,6... the remainders form the sequence 0,0,1,0,0,4...

These sequences are not the same, however.
Find the smallest number that gives a different remainder upon dividing these two numbers.

Feel free to find more.