Number Theory

Computer Science Level 3

Consider $2^5 = 32$ , where the sum of the digits $3+2 = 5$ , which is equal to the power of 2.

Which other $n$ , such that the sum of digits of $2^n$ equal to $n$ ?

The answer is 70.

1 solution

Giorgio Coniglio
Aug 7, 2016

2^70 = 1180591620717411303424 where the sum of the digits is 70.

Are these two the only solutions? Is there a proof for it?

Anand Chitrao - 4 years, 10 months ago

What about 32? 2^5 and sum is 5

דניאל נוקראי - 4 years, 10 months ago

Yes, you are right, I need to correct the problem. In the example I gave was for n > 10 which were not mentioned.

Giorgio Coniglio - 4 years, 10 months ago

Isn't their some method?Do we have to solve this just through hit and trial?I think that would take a lot of time.

Anandmay Patel - 4 years, 10 months ago

@Anandmay Patel – The first time I came across this problem I was wondering if I could mount a system of equations with the sum of the digits equal to n, but it was clear that it wouldn't work since the number of variables would change with n. As such, I tried to find the upper limit of n and I checked for 2^100 and got the sum of digits 115. Then I checked for 2^200 and got 256. So I realized that n would be less than 100 and I worked backwards and found easily n=70.

Giorgio Coniglio - 4 years, 10 months ago

Number Theory

The answer is 70.

1 solution

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