What 4-digit number abcd satisfies this equation?

4 * abcd = dcba (Worried)

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- Jun 8th 2010, 01:41 AMchinchuWhat 4-digit number abcd satisfies this equation?
What 4-digit number abcd satisfies this equation?

4 * abcd = dcba (Worried) - Jun 8th 2010, 02:55 AMsimplependulum
Since so but if from the unit digit of we conclude that is odd but it is the multiple of which should be even . Therefore ,

Consider the unit digit after multiplying a number among by , we have already known it is

we have but is impossible because we conclude

Then consider the actual values of these number :

we have

When

When it is impossible , so is the unique solution .

The required number is

is an indeterminated equation , any number may be the solution of provided will be an integer :

because is an integer , must be a positive odd , but if is very large which makes to be greater than , we cannot construct a number with a digit ... - Jun 8th 2010, 03:19 AMchinchuhi
how did u take b =1 and b = 3??

- Jun 8th 2010, 04:58 AMdarknight
13b + 1 = 2c

c = (13b + 1)/2

because is an integer , must be a positive odd since for (b + 1) to be perfectly divisible by 2 it should be even. And if b+1 is even it follows that b is odd.

And therefore simplependulum started by trying all the odd integers which start from 1. And quite clearly, the next odd number is 3. Putting b as 3 results in c = 20. Since c lies between 0 and 9, b cannot be 3 or any integer greater than 3. Therefore, b = 1. - Jun 8th 2010, 06:36 AMundefined
Initially I misread dcba as dabc and thought there were no solutions. See this Wikipedia article for some interesting related material.

- Jun 8th 2010, 06:57 AMdarknight
That was very interesting. I only knew about 142857 and it's connection to 22/7. Feels good figuring out its intricacies.

- Jun 8th 2010, 07:50 AMSoroban
Hello, chinchu!

This is a classic puzzle.

It can be solved without Algebra, just common sense.

Quote:

What 4-digit number satisfies: . ?

We have:

. .

In column-1, since is a digit ,

. . we see that

But if , column-4 says: . ends in 1 . . . impossible.

Hence: .

. .

In column-1, we see that: .

In column-4, ends in 2.

. . Hence: .

. .

We see that there is no "carry" from column-2.

. . Hence: .

. . Suppose

. . . .

. . In column-3, we have: . ends in 0 . . . impossible.

Hence: .

. .

In column-3, ends in 1.

. . Hence: .

Since

Solution:

. .

- Jun 19th 2010, 09:59 PMshannu82
why does it say LaTeX ERROR: Unknown error all over the place.

- Jun 19th 2010, 10:06 PMsimplependulum