You can do it , I mean find it

This is definitely not the most elegant solution but it worked well for me. Start by creating constraints for the problem.

Since the solution must still have 6 digits when multiplied by 6, the solution must lie on the interval 100,000 <= x <= 166,666. (Because 166,666 is the integer immediately less than 1,000,000/6)

Given this, we are able to deduce that the first digit of our solution, x, must be 1.

We then proceed to find values or 'critical points' at which our multipliers (i.e. 2, 3, 4, 5, & 6) move up to the next block of 100,000. For example, a critical point for our 2 multiplier would be 150,000, because 150,000 x 2 = 300,000. These critical numbers will be used to create our test intervals.

Critical Points (between 100,000 and 166,666)

2: 150,000

3: 133,334

4: 125,000, 150,000

5: 120,000, 140,000, 160,000

6: 116,667, 133,334, 150,000

Test Intervals: (labeled with letters for future reference)

A: 100,000 - 116,666

B: 116,667 - 119,999

C: 120,000 - 124,999

D: 125,000 - 133,333

E: 133,334 - 139,999

F: 140,000 - 149,999

G: 150,000 - 159,999

H: 160,000 - 166,666

By taking any number within or on the boundary of our test intervals and multiplying it by 2, 3, 4, 5, & 6, we are given a unique 'digit' in the 100,000's place of our new number. We know this 'digit' must appear in the original number as well. Note that '1' appears in all of them, as it appears in the 100,000's place for all possible solutions.

Given this fact, we list all digits that must appear in the solution (unique to each test interval):

A: 1, 2, 3, 4, 5, 6

B: 1, 2, 3, 4, 5, 7

C: 1, 2, 3, 4, 6, 7

D: 1, 2, 3, 5, 6, 7

E: 1, 2, 3, 4, 5, 6, 8

F: 1, 2, 4, 5, 7, 8

G: 1, 3, 4, 5, 6, 7, 9

H: 1, 3, 4, 6, 8, 9

Each interval has at least 6 required digits, so no digits are able to be repeated. Therefore, the following intervals can be immediately discarded:

A: (requires 0's or repeated 1's)

B: (requires repeated 1's)

E: (more required digits than available spaces)

G: (more required digits than available spaces)

The four remaining intervals may be revised as follows:

C: 123,467 - 124,763

D: 125,367 - 132,765

F: 142,578 - 148,752

H: 163,489 - 164,983

Before testing all possible permutations, many potential solutions can be immediately discarded based on their final digit:

0 - no permutation will include this as its final digit

1 - no permutation will include this as its final digit (1 will always be in the 100,000's place, never in the units place)

2, 4, 6, 8 - when multiplied by the 5 multiplier, the solution would contain a 0.

5 - when multiplied by the 2, 4 or 6 multiplier, the solution would contain a 0.

This leaves only 3, 7, and 9 as acceptable final digits for our permutations. Furthermore, these final digits carry conditional stipulations and, at times, can also be discarded:

3 - Must appear in an interval along with 6, 9, 2, 5, and 8. (No such interval exists.)

7 - Must appear in an interval along with 4, 1, 8, 5, and 2. (Only Interval F satisfies this.)

9 - Must appear in an interval along with 8, 7, 6, 5, and 4. (No such interval exists.)

Therefore we can conclude that our solution must be found in Interval F and have a 7 as its final digit.

The only permutations that satisfy these criteria are: 142,587, 142,857, 145,287, 145,827, 148,257, 148,527.

Testing each of these manually against our multipliers, we find that x = 142,857.