Find the Missing Digit in 6?22: Divisibility by 9 Problem

Complete the number so that it is divisible by 9 without a remainder:

$6?22$

To solve this problem, we'll use the divisibility rule for 9, which states that a number is divisible by 9 if the sum of its digits is divisible by 9.

• Step 1: Calculate the sum of the known digits.
  The number given is $6?22$, which means we have digits 6, ?, 2, and 2.
  Calculating the sum of known digits: $6 + 2 + 2 = 10$.
• Step 2: Add the missing digit $?$ and check for divisibility by 9.
  The total sum considering the missing digit will be $10 + ?$.
• Step 3: Determine which value of $?$ makes $10 + ?$ divisible by 9.
  We analyze the expression $10 + ? \equiv 0 \pmod{9}$.

Trying each option:

• If $? = 0$, $10 + 0 = 10$ (not divisible by 9).
• If $? = 1$, $10 + 1 = 11$ (not divisible by 9).
• If $? = 2$, $10 + 2 = 12$ (not divisible by 9).
• If $? = 3$, $10 + 3 = 13$ (not divisible by 9).
• If $? = 4$, $10 + 4 = 14$ (not divisible by 9).
• If $? = 5$, $10 + 5 = 15$ (not divisible by 9).
• If $? = 6$, $10 + 6 = 16$ (not divisible by 9).
• If $? = 7$, $10 + 7 = 17$ (not divisible by 9).
• If $? = 8$, $10 + 8 = 18$ (divisible by 9).
• If $? = 9$, $10 + 9 = 19$ (not divisible by 9).

Therefore, the correct digit to place in the question mark to make the number divisible by 9 is $8$.