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Missing Digit II (Posted on 2015-03-10)

Given that the base ten representation of 2²⁹ has precisely 9 distinct digits – determine, without direct evaluation, the missing digit.

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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Solution

| Comment 1 of 5

29*log(2) = 8.72987, then 2^29 has 9 digits total. KS has told us there are 9 distinct digits in its representation. So each of nine digits occurs exactly once in 2^29

The sum of all ten digits mod 9 equals 0.

The (sum of the digits of 2^29) mod 9 will equal 2^29 mod 9.

2^29 mod 9 = ((2^6)^4)*(2^5) mod 9 = (64^4)*32 mod 9 = (1^4)*5 mod 9 = 5.

5 + 4 = 0 mod 9. Therefore 4 must be the missing digit.

Checking by direct calculation 2^29 = 536870912

Posted by Brian Smith on 2016-06-26 11:33:57

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