The missing digit of two to the twenty-ninth

Question

The number 2 to the power 29 has nine digits, all different. Which of the ten digits 0 to 9 does it not contain? You may not compute 2 to the 29 (it is 536,870,912, but pretend you do not know that); find the missing digit by reasoning.

Accepted Answer

The missing digit is 4. Digit sums modulo 9. Any number and the sum of its digits leave the same remainder on division by 9. If 2 to the 29 uses nine different digits, its digit sum is 45 minus (the missing digit), because all ten digits add to 45. Remainder of 2 to the 29 modulo 9. Powers of 2 modulo 9 cycle: 2, 4, 8, 7, 5, 1, and then repeat with period 6 (since 2 to the 6 = 64 = 63 + 1). Now 29 = 4 × 6 + 5, so 2 to the 29 has the same remainder as 2 to the 5 = 32, which is 5 modulo 9. Solve. The digit sum 45 minus m must leave remainder 5 when divided by 9. Since 45 is a multiple of 9, thi…

The missing digit of two to the twenty-ninth

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