Largest prime number with all digits different

Question

What is the largest prime with distinct digits? (It is certainly less than ten digits long.Can you explain it why?

Answer 1

The answer should be $$p=987654103.$$ As any number using all ten digits would by a multiple of $3$, we are left with only few nine-digit candidates $987654xyz$ that can be checked manually.

Answer 2

Hint: Suppose you had all ten digits - what would the sum of the digits be?

Answer 3

This question depends on the number base being used. It is assumed that the OP meant base 10, but there is nothing special about that (or the answer) except that humans generally have ten fingers. Here are a few others in different bases:

• base 2: 10 (2)
• base 3: 201 (19)
• base 4: 103 (19, using all three non-zero digits always results in a number divisible by 3)
• base 5: 4302 (577, using all four non-zero digits always results in a number divisible by 2)

Higher bases are left as an excerise for the reader!

Answer 4

If all the digits are used, it will be divisible by 3, right? Same also if you will use the digits 1-9. Therefore, the largest prime number with different digits is less than 10 digits. If it would be nine digits, O is included as one of the digits.