The Missing Digit

The number 2²⁹ has 9 digits, all different. Which digit is missing?

Bonus: Is the number 9991 prime?

Let the missing digit be m. Every number and the sum of its digits give the same remainder when divided by 9. The number 2²⁹ = 32 * 64⁴ gives remainder 5 when divided by 9, and therefore 9 divides (0 + 1 + 2 + … + 9) – 5 – m = 40 – m. Thus, the missing digit is 4.

Bonus: 9991 = 10000 – 9 = 100² – 32 = (100 – 3)(100 + 3) = 97 * 103. Therefore the number 9991 is not prime.

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