What is the largest integer n with no repeated digits that is relatively prime to 6?

Note that two numbers are considered relatively prime if they share no common factors besides 1.

Answer: 987654301

Step-by-step explanation:

The factors of 6 are 2 and 3:

2*3 = 6.

So we want a big number that is no divisible by 2 or 3.

Then:

The number can not end in 0, 2, 4, 6 or 8.

The addition of the digits can not add to a multiple of 3.

If we want to construct the largest number possible, we should try to use the largest number at the beginning, so we have:

9876543210

but we can not end in 0, so we can write this as:

9876543201

but 9 + 8 + 7 + 6 + 5 + 3 + 2 + 1 = 45, it is a multiple of 3, so we should discard one of these numbers, lets discard the 1, as it is the smaller one.

we can write:

987654203 but if instead we remove the 2, we can leave the 3 in the place and get:

987654301 that is bigger, it ends with an odd number so it is not a multiple of 2, and the addition of all the digits adds up to 44, so this is not a multiple of 3.

Then this is the answer,