Help to Find the smallest natural number with these features:

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February 19th 2013, 06:13 AM
Sott

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 [ http://latex.codecogs.com/gif.latex?...n%20\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.

(Thinking)
February 19th 2013, 06:26 AM
Kmath

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
February 19th 2013, 06:41 AM
Kmath

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

Also,
the second digit can not be 9,8,7,6,5
February 19th 2013, 06:59 AM
Sott

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

Why the original number should end with 7? :)
February 19th 2013, 07:13 AM
Kmath

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.
February 19th 2013, 08:03 AM
Sott

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 :/
February 19th 2013, 08:13 AM
Sott

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

So how to solve it? (Worried)
February 19th 2013, 09:20 AM
Soroban

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

Hello, Sott!

Quote:

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

We see that $E = 7.$

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

We see that $D = 5.$

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

We see that $C = 8.$

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

We see that $B = 2.$

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

We see that $A = 4.$

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

The smallest such number is $142857.$
February 19th 2013, 09:33 AM
Sott

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

WOW! Awesome Soroban! Thank you very much!!! (Happy)
February 19th 2013, 10:54 AM
Kmath

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

Quote:

Originally Posted by Soroban

Hello, Sott!

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

We see that $E = 7.$

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

We see that $D = 5.$

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

We see that $C = 8.$

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

We see that $B = 2.$

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

We see that $A = 4.$

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

The smallest such number is $142857.$

Really nice work, but I think you have first to check it for the 3,4 and 5 digits