Question: What is the smallest four digit multiple of 9 with four distinct digits?

It looks easy but I don't want my solution to be found just through trial and error, is there a better and more efficient way to do this problem without just trying all multiples of nine? Thanks!! 🙏

RoseLatte Aug 3, 2023

#1**+3 **

*What is the smallest four digit multiple of 9 with four distinct digits?*

I did a little bit of logic and a little bit of brute force.

We want the lowest four digit number. Well, it has to start with a 1. So we've got that.

What second digit would be the lowest. Obviously, 0. So we've got 10 _ _.

For the third digit, it can't be 0 or 1 so let's go with 2. Now we have 102 _.

The fourth digit can't be 0 or 1 or 2. It has to be 3 or larger.

That's the end of the logic. Now for the brute force.

Tried 1023, not evenly divisible by 9.

Tried 1024, not evenly divisible by 9.

Tried 1025, not evenly divisible by 9.

Tried 1026, **Bingo**. It's 114. So that's your number right there: **1026**.

_{.}

Bosco Aug 3, 2023

#1**+3 **

Best Answer

*What is the smallest four digit multiple of 9 with four distinct digits?*

I did a little bit of logic and a little bit of brute force.

We want the lowest four digit number. Well, it has to start with a 1. So we've got that.

What second digit would be the lowest. Obviously, 0. So we've got 10 _ _.

For the third digit, it can't be 0 or 1 so let's go with 2. Now we have 102 _.

The fourth digit can't be 0 or 1 or 2. It has to be 3 or larger.

That's the end of the logic. Now for the brute force.

Tried 1023, not evenly divisible by 9.

Tried 1024, not evenly divisible by 9.

Tried 1025, not evenly divisible by 9.

Tried 1026, **Bingo**. It's 114. So that's your number right there: **1026**.

_{.}

Bosco Aug 3, 2023

#2**+2 **

Here's an approach that only uses a little trial and error

Note that 999/9 = 111

So....let's assume that our number must have the form 1 0 __ ___ where the last two digits are unique

Note that successive multiples of 9 (after 999) are

1008

1017

1026 = our number

CPhill Aug 3, 2023