In a $4$-digit positive number, use algebra to show that if the sum of all $4$ digits is divisib

Question

In a $4$-digit positive number, use algebra to show that if the sum of all $4$ digits is divisible by $3$, then the number is also divisible by $3$. Explain if this only applies to $4$-digit positive numbers.

Accepted Answer

(a) $N=3(333a+33b+3c)+(a+b+c+d)$, so if the digit sum is divisible by $3$, then $N$ is divisible by $3$. (b) This applies to every positive integer because $10^k-1$ is divisible by $3$ for all non-negative integers $k$.

Solving Linear Equations

Solving Linear Equations

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