DivideByNineMethod Basic

Problem - 311
The number $2^{29}$ is a nine-digit number whose digits are all distinct. Which digit of $0$ to $9$ does not appear?

Answer     4

$\textbf{Answer}$

The digit $\boxed{4}$ is missing.

$\textbf{Solution}$

This problem involves a math theorem which is often referred as the MOD by $9$ rule. This rule states that the remainder of any given positive integer when being divided by $9$ is the same as the remainder when the sum of its digits is divided by $9$. For example, $123$ divides $9$ has a remainder of $6$. This is the same as the remainder of $1+2+3=6$ divides $9$. 

By the MOD by $9$ technique, the sum of these digits must be congruent to $2^{29}\pmod{9}$ which is $(-4)$ (see below). Hence, the missing digit is $\boxed{4}$. $$2^{29} \equiv (2^3)^9 \times 2^2 \equiv (-1)^9\times 4 \equiv -4\pmod{9}$$

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