In a $3$-digit positive number, use algebra to show that if the sum of all $3$ digits is divisib

Question

In a $3$-digit positive number, use algebra to show that if the sum of all $3$ digits is divisible by $9$, then the number is also divisible by $9$. Explain if this only applies to $3$-digit positive numbers.

Accepted Answer

(a) $N=9(11a+b)+(a+b+c)$, so if the digit sum is divisible by $9$, then $N$ is divisible by $9$. (b) This applies to every positive integer because $10^k-1$ is divisible by $9$ for all non-negative integers $k$.

Solving Linear Equations

Solving Linear Equations

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