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RMO 2014 Delhi Region Q.3

Suppose for some positive integers $$r$$ and $$s$$, the digits of $$2^r$$ is obtained by permuting the digits of $$2^s$$ in decimal expansion. Prove that $$r=s$$

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Note by Aneesh Kundu
2 years, 5 months ago

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This one went straight like a tangent over my head. · 2 years, 5 months ago

if r < or = s 2^s - 2^r = 2^r(2^n - 1) where r+n=s 9 divides 2^n-1 (2^r is a permutation of the digits of 2^s) This is only possible when 2^n=1 Therefore, n=0 which implies that r=s · 1 year, 5 months ago

I have also not solved this. But if we assume the number of digits of 2^r and 2^s are same here is a simple solution, · 2 years, 5 months ago

Without loss of generality let us assume r>s. Let r=s+k. where k>0 As 2^r and 2^s have same number of digits k<4. [ Since k>=4 will imply multiplying 2^s by a number >10 and thus ending up with a greater number of digits.] now as they have same digits their sum of digits is same SO they are congruent modulo 9. So 2^s+k - 2^s = 2^s( 2^k -1) ==0 (mod 9). Since (9,2^s)=1(i.e they are co-prime) this implies 2^k -1==0(mod 9). k<4 implies 2^k -1 can take values 1 , 3 or 7 all of which leads to a contradiction downright. So k=0 giving r=s. · 2 years, 5 months ago

Can you explain why you chose to use modulo 9 rather than modulo 3? Using modulo 3 allows k = 2 to work. · 2 years, 5 months ago

What do you mean? Modulo 3 will fail to contradict the case k=2 so modulo 9 is the only option left. · 2 years, 5 months ago

So one would have to first see that modulo 3 fails before you know to use modulo 9? · 2 years, 5 months ago

Why $$k\leq 4$$

Aren't we just rearranging the digits? · 2 years, 5 months ago

Of course!Permutation means rearrangement. k>=4 means we have to multiply 2^s by 16 or more which is >10.An that will lead us to more number of digits in 2^r · 2 years, 5 months ago

oh..sorry

Actually I misread $$r=s+k$$ as $$r+s=k$$, thats why I was wondering that $$k<4$$ could be a really useful result. · 2 years, 5 months ago

Apparantly this method fails when k=6n as 2^6n==1 (mod 9) which does not provide a contradiction. But this case can surely arise when THERE ARE A NUMBER OF ZEROES AS DIGITS IN 2^r. · 2 years, 5 months ago

Please anybody provide solution for this part · 2 years, 5 months ago

if $$k=6n$$ and $$k<4$$, then $$\dfrac{2}{3}>n$$ this leaves us with no positive integer values for $$n$$. · 2 years, 5 months ago

Noooo! We get k<4 only when we know that the number of digits in 2^r and 2^s are same. But if suppose there is a zero in 2^r. When we permute the digits the zero can bemade to come in front thus reducing the number of digits in 2^s. And then we cant say k<4. · 2 years, 5 months ago

Comment deleted Dec 08, 2014

I have also done the same sums. Do u think it is enough?? @ very anxious · 2 years, 5 months ago

what marks would you get to show that r-s<4 in this question < i could only proceed till here> · 2 years, 4 months ago

Suppose s ≤ r. If s < r then 2s < 2 r . Since the number of digits in 2s and 2r are the same, we have 2r < 10 × 2 s < 2 s+4. Thus we have 2s < 2 r < 2 s+4 which gives r = s + 1 or s + 2 or s + 3. Since 2r is obtained from 2s by permuting its digits, 2r − 2 s is divisible by 9. If r = s + 1, we see that 2r − 2 s = 2s and it is clearly not divisible by 9. Similarly, 2s+2 − 2 s = 3 × 2 s and 2s+3 − 2 s = 7 × 2 s and none of these is divisible by 9. We conclude that s < r is not possible. Hence r = s · 2 years, 5 months ago

What exactly is the question? Are leading 0's allowed or not? · 2 years, 5 months ago

I don't get it · 2 years, 5 months ago

Since its not mentioned in the question whether both the power have the same no digits or not, we need to construct a general case.

Its strange that they still haven't uploaded the official solutions till now. · 2 years, 5 months ago

That is the precise problem.Nothing is mentioned about the leading zeroes. I think if we consider the leading zeroes the standard of the question goes well above RMO. But i was stuck at this very juncture and failed to provide the above mod 9 solution. · 2 years, 5 months ago

Thee official key says the number of digits are same.I was trying to solve the more general case.I couldnt get it.But i noticed a strange thing 2^{34}=134217728 and 2^{30}=1073741824 and they have almost same digits that is they differ just 1 digit (2 and 4) I don`t think this may help But is it true? Thanks · 2 years, 5 months ago

Actually i should have felt that the general case is too difficult! the sad part is i felt it after submitting the answer script. Actually this means nothing now but still it was too much of spoilt milk not to cry over!! Jokes apart your observation is really interesting but i heard recently that someone have proved it,( though i haven't seen it myself) · 2 years, 5 months ago

i did not enjoy this problem at all! :( · 2 years, 5 months ago

Comment deleted 5 months ago