Flexible Powers of 2

Number Theory Level 3

We define a perfect power of 2 to be flexible if we can rearrange the digits in the number to produce a distinct perfect power of 2. Find the total number of perfect power of 2 that are flexible .

Note: Leading zeros are not allowed.


The answer is 0.

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Harish Nandan by Harish Nandan , 21, India

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1 solution

Harish Nandan
Feb 11, 2016
  • First of all according to question Number of digits should be same .
  • Now if 2^n and 2^(n+k) are two numbers and they have same number of digits then k has to be 1,2 or 3 . Cause if you take k to be 4 or more then 2^n+k = 2^4 * 2^n.Since 2 to the power FOUR is 16 that is more than 10 , hence numbers of digits will increase.
  • Sum of digits are same So on division by 9 both of them should leave the same remainder.
  • If you chose any arbitrary 2^n Then only possibilities are 2^n+k where k is 1,2 or 3. Now you check the periodicity of remainders of 2^n on division by 9 ...no repetition is there for four consecutive powers.Therefore no such power is possible.
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