# Thread: Help to Find the smallest natural number with these features:

1. ## Help to Find the smallest natural number with these features:

I understand that I must learn math myself, but this exercise isn't as easy as I thought, so I need a help or any advices how to solve it.

So, we have to find the smallest natural [ $\dpi{150} x \varepsilon \mathbb{N}$ ] number with these features:

1) The first digit is 1.

2) If we moved 1 to the end of the number, we would get a three times bigger/greater number.

2. ## Re: Help to Find the smallest natural number with these features:

the first digit is 1,
since 3 times the number ends with 1 the original number should end with 7
also the number is divisible by 3.
so you have to check numbers such that : the f number starts with one, ends with seven and divisable by 3

3. ## Re: Help to Find the smallest natural number with these features:

Also,
the second digit can not be 9,8,7,6,5

4. ## Re: Help to Find the smallest natural number with these features:

Why the original number should end with 7?

5. ## Re: Help to Find the smallest natural number with these features:

since when you multiply it by three it should end with 1. If you tried other possibility you would never get 1.

6. ## Re: Help to Find the smallest natural number with these features:

I know that number can be divisable by 3 if the sum of number's digits is divisable by 3. I tried to find an answer, but I think that I am doing wrong something. I think that number is really big :/ I checked some four-figure numbers, but all of them were wrong. Hmmm :/

7. ## Re: Help to Find the smallest natural number with these features:

So how to solve it?

8. ## Re: Help to Find the smallest natural number with these features:

Hello, Sott!

Find the smallest natural number with these features:
. . 1) The first digit is 1.
. . 2) If we moved 1 to the end of the number, we would get a three times greater number.

We have: .$\displaystyle \begin{array}{cccccc}1&A&B&C&D & E \\ \times &&&&& 3 \\ \hline A&B&C&D& E & 1 \end{array}$

We see that $\displaystyle E = 7.$

We have: .$\displaystyle \begin{array}{cccccc}1&A&B&C&D&7 \\ \times &&&&& 3 \\ \hline A&B&C&D& 7 & 1 \end{array}$

We see that $\displaystyle D = 5.$

We have: .$\displaystyle \begin{array}{cccccc}1&A&B&C&5&7 \\ \times &&&&& 3 \\ \hline A&B&C&5& 7 & 1 \end{array}$

We see that $\displaystyle C = 8.$

We have: .$\displaystyle \begin{array}{cccccc}1&A&B&8&5&7 \\ \times &&&&& 3 \\ \hline A&B&8&5& 7 & 1 \end{array}$

We see that $\displaystyle B = 2.$

We have: .$\displaystyle \begin{array}{cccccc}1&A&2&8&5&7 \\ \times &&&&& 3 \\ \hline A&2&8&5& 7 & 1 \end{array}$

We see that $\displaystyle A = 4.$

We have: .$\displaystyle \begin{array}{cccccc}1&4&2&8&5&7 \\ \times &&&&& 3 \\ \hline 4&2&8&5& 7 & 1 \end{array}$

The smallest such number is $\displaystyle 142857.$

9. ## Re: Help to Find the smallest natural number with these features:

WOW! Awesome Soroban! Thank you very much!!!

10. ## Re: Help to Find the smallest natural number with these features:

Originally Posted by Soroban
Hello, Sott!

We have: .$\displaystyle \begin{array}{cccccc}1&A&B&C&D & E \\ \times &&&&& 3 \\ \hline A&B&C&D& E & 1 \end{array}$

We see that $\displaystyle E = 7.$

We have: .$\displaystyle \begin{array}{cccccc}1&A&B&C&D&7 \\ \times &&&&& 3 \\ \hline A&B&C&D& 7 & 1 \end{array}$

We see that $\displaystyle D = 5.$

We have: .$\displaystyle \begin{array}{cccccc}1&A&B&C&5&7 \\ \times &&&&& 3 \\ \hline A&B&C&5& 7 & 1 \end{array}$

We see that $\displaystyle C = 8.$

We have: .$\displaystyle \begin{array}{cccccc}1&A&B&8&5&7 \\ \times &&&&& 3 \\ \hline A&B&8&5& 7 & 1 \end{array}$

We see that $\displaystyle B = 2.$

We have: .$\displaystyle \begin{array}{cccccc}1&A&2&8&5&7 \\ \times &&&&& 3 \\ \hline A&2&8&5& 7 & 1 \end{array}$

We see that $\displaystyle A = 4.$

We have: .$\displaystyle \begin{array}{cccccc}1&4&2&8&5&7 \\ \times &&&&& 3 \\ \hline 4&2&8&5& 7 & 1 \end{array}$

The smallest such number is $\displaystyle 142857.$
Really nice work, but I think you have first to check it for the 3,4 and 5 digits