A number theory problem by Syed Baqir

Number Theory Level 5

Can you find the smallest positive number such that if you shuffle the digits of the number in a particular order, the shuffled number becomes twice the original number.

The answer is 125874.

2 solutions

Shivam Yaduvanshi
Jul 9, 2015

Expecting it to be a 2 digit no. taking 2 (10a + b) = 10b + a gives 19a=ut 8b now d taking L.C.M of 19 and 8 ............ to find later that condition is not satisfied and continuing to 3 digit and goin on ...............................................

but is a very long process .... does any1 hav a shorter way by d way ididn't continued i googled

Syed Baqir
Jul 5, 2015

125874 => 251748

251748 is twice the 125874 and have same digits 1,2,4,5,7 & 8

How can we show that this is indeed the smallest such number?

Calvin Lin Staff - 5 years, 11 months ago

I cant see any method to prove that it is smallest positive number .

Syed Baqir - 5 years, 11 months ago

I think that would be the interesting part. It is not clear to me how to even arrive at this solution in the first place (other than through brute force computation).

For example, another solution (ignoring smallest) would be $142857 \times 2 = 285714$ , which would be the answer that I expect most people to give.

Calvin Lin Staff - 5 years, 11 months ago

The Mathematica function

Test[n_] := Sort[IntegerDigits[2n]] == Sort[IntegerDigits[n]]

identifies if a number $n$ can have its digits reordered to form its double. The command

Select[Range[130000],Test[#] &]

gives the answer

125874, 128574

which seems to settle the matter, if in an unsatisfying manner!

Mark Hennings - 3 years, 8 months ago

Yea, I don't think there is a nice mathematical proof to this.

Calvin Lin Staff - 3 years, 8 months ago

This is particularly interesting since these are the recurring digits from division by 7

Stephen Mellor - 3 years, 8 months ago

Another person posted this problem, and gave the answer 142857. But, clearly, your answer is smaller. 1/7 vs. 18/143?

James Wilson - 3 years, 7 months ago

If you look at my reply to Calvin, there are two 6-digit answers possible less than 130000, and the one here is the smaller of the two. There are doubtless even larger ones.

Mark Hennings - 3 years, 7 months ago

Yes, I know. I saw it. I just commented to get the 18/143 in there somewhere.

James Wilson - 3 years, 7 months ago

@James Wilson – And to let you know someone else posted this question with the incorrect answer.

James Wilson - 3 years, 7 months ago

A number theory problem by Syed Baqir

The answer is 125874.

2 solutions

0 pending reports