If the number $3422213pq$ is divisible by 99, find the missing digits $p$ and $q$.

### Concept & Formula To check divisibility by 99, we use its co-prime factors, 9 and 11. $$ 99 = 9 \times 11 $$ The number must satisfy both divisibility rules: 1. Sum of digits must be a multiple of 9. 2. The difference between the sum of alternating digits must be 0 or a multiple of 11. ### Step-by-Step Solution - **Given:** The number $3422213pq$ is divisible by 99. - **Calculation (Divisibility by 9):** Sum of digits: $3 + 4 + 2 + 2 + 2 + 1 + 3 + p + q = 17 + p + q$. For this to be a multiple of 9, $17 + p + q$ can be 18 or 27. This simplifies to $p + q = 1$ (Eq. i) or $p + q = 10$ (Eq. ii). - **Calculation (Divisibility by 11):** Sum of odd places: $q + 3 + 2 + 2 + 3 = q + 10$ Sum of even places: $p + 1 + 2 + 4 = p + 7$ Difference: $(q + 10) - (p + 7) = q - p + 3$ For this to be 0 or 11: $q - p = -3 \implies p - q = 3$ (Eq. iii) OR $q - p + 3 = 11 \implies q - p = 8$ (Eq. iv) - **Deduction:** If $p + q = 1$, it does not yield single-digit integer solutions with (iii) or (iv). Solving $p + q = 10$ and $q - p = 8$: Adding both equations gives $2q = 18 \implies q = 9$. Substituting $q = 9$ into $p + q = 10$ gives $p = 1$. ### Exam Strategy & Shortcut Use option elimination! Instead of solving simultaneous equations, test the given options using the rule of 9. The sum of known digits is 17. Testing $p=1, q=9$: Sum = $17 + 1 + 9 = 27$ (Divisible by 9). Check the alternating difference for 11: $(9+10) - (1+7) = 19 - 8 = 11$ (Divisible by 11). Both rules pass instantly. ### Common Pitfall Setting up the alternating difference equation backwards or miscalculating basic addition when combining variables and constants. Always strictly group the alternating places before finding the difference to avoid sign errors. ### Final Answer Therefore, the correct answer is 1, 9.