choose 4 numbers from 1 to 5 to form a 4-digit number. What is the biggest such 4-digit number divisible by 6?

For a 4-digit number which is divisible by 3, the sum of all digits are divisible by 3
1 + 2 + 3 + 4 + 5 = 15 is divisible by 3.
If the digit 3 is not chosen, the sum is also divisible by 3, i.e.
1 + 2 + 4 + 5 = 12 is divisible by 3.
Therefore, the digits 1, 2, 4 and 5 are chosen to form a 4-digit number.

As the 4-digit number is even, the units digit must be 4 or 2.
To make a largest number, the largest digit should be put into the leftmost digit, and then and second largest digit, and so on.

Hence, the required 4-digit number = 5412

Assuming the 4 numbers/digits are distinct, the answer is 5412.

A number is divisible by 6 only if it is divisible by 2 and by 3.
This tells us that the number is even with the sum of its digits divisible by 3.
This is possible only if the 4 numbers/digits 1,2,4,5 are used.
The largest number in question that can be formed from these 4 digits is exactly 5412.

5412

To be divisible by 6, first it must be even; second, it must also be divisible by 3.

If you don't repeat any numbers, the largest possible 4-digit number is 5,412 (NOT 5,421; it's not even, so it can't be divisible by 6, and NOT 5,432; sum of digits is 14, which isn't divisible by 3, so the whole number isn't divisible by 6).

If you decide to repeat numbers, then 5,544.

　
You don't mention if repeated digits are allowed.

Digits: 1, 2, 3, 4, 5

A number is divisible by 6 if it is divisible by 2 and 3, i.e.
• number ends with even digit (2 or 4)
• sum of digits is divisible by 3

——————————————————————————————

For repeats, the 2 largest even numbers are:
5554 ----> sum of digits = 19
5552 ----> sum of digits = 17

To make 1st number divisible by 3, subtract 1 from a digit
To make 2nd number divisible by 3, subtract 2 from a digit, or subtract 1 from 2 digits
5544, 5532

The largest number divisible by 6 is 5544

——————————————————————————————

For numbers with no repeated digits:

Sum of digits from 1 to 5 = 1 + 2 + 3 + 4 + 5 = 15

Now we must pick 4 digits whose sum is divisible by 3, i.e.
we must eliminate a digit (n) so that 15-n is divisible by 3
n = 3

Therefore, the 4 digits that will give a number divisible by 3 are:
1, 2, 4, 5

Number must be divisible by 2, so it ends in 2 or 4.

The largest such number is: 5412

If each digit is different, the biggest such numbers are:
5555: not a multiple of 6
5554: not a multiple of 6
5553: not a multiple of 6
5552: not a multiple of 6
5551: not a multiple of 6
5545: not a multiple of 6
5544: this is the answer

Or if you want to choose 4 DIFFERENT numbers from 1 to 5, the biggest such numbers are:
5432: not a multiple of 6
5431: not a multiple of 6
5423: not a multiple of 6
5421: not a multiple of 6
5413: not a multiple of 6
5412: this is the answer