## Project Euler 52: Find the smallest positive integer, x, such that 2x, 3x, 4x, 5x, and 6x, contain the same digits in some order

#### Problem Description

It can be seen that the number, 125874, and its double, 251748, contain exactly the same digits, but in a different order.

Find the smallest positive integer, x, such that 2x, 3x, 4x, 5x, and 6x, contain the same digits.

#### Analysis

This is a rehash of an old high school problem on repeating digits:

1/7 = 0.142857 142857 142857 ...

2/7 = 0.285714 285714 285714 ...

3/7 = 0.428571 428571 428571 ...

4/7 = 0.571428 571428 571428 ...

5/7 = 0.714285 714285 714285 ...

6/7 = 0.857142 857142 857142 ...

I’m sure you get the idea without giving away the answer.

As for our programmatic solution: an assumption that the digits didn’t have to be unique, I set the number of digits less one as our starting bound and search sequentially by sorting the digits within each number and comparing for multiples 1, 2, 3, 4, 5 and 6.

We can begin our search with a 5 digit multiple of 9, such as 99999, because the result has at least 6 digits. Also, because any number and its permutation always differ by a multiple of 9 (ex., (45121-11542)/9 = 3731) we can increment by 9 instead of 1.

#### Project Euler 52 Solution

Runs < 0.015 seconds in Python 2.7.Use this link to get the Project Euler 52 Solution Python 2.7 source.

#### Answer

Slowly swipe from either end beginning with the white vertical bar to get an idea of the starting or ending digits. For less drama, just double click the answer area. The distance between the two bars will give you an idea of the magnitude. Touch devices can tap and hold the center of the box between the two bars and choose*define*to reveal the answer.

#### Afterthoughts

- The next one found was 1428570

*Project Euler 52 Solution last updated*

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