The digits indicated by $*$ in $3422213**$ so that this number is divisible by 99 are

## Concept & Logic For a number to be perfectly divisible by $99$, it must pass the divisibility tests for both of its co-prime factors: **$9$ and $11$**. * **Rule of 9:** The sum of all digits must be a multiple of 9. * **Rule of 11:** The alternating sum of the digits must yield 0 or a multiple of 11. ## Step-by-Step Solution **Given:** The number is $3422213ab$, where $a$ and $b$ represent the missing digits in that exact order (tens, units). **Calculation / Deduction:** * **Step 1: Check the Rule of 9.** Sum of given digits: $3 + 4 + 2 + 2 + 2 + 1 + 3 = 17$. Total sum $= 17 + a + b$. Let's look at the given options for $(a, b)$ and their sums: * (a) $1 + 9 = 10 \implies$ Total sum $= 27$ * (b) $3 + 7 = 10 \implies$ Total sum $= 27$ * (c) $4 + 6 = 10 \implies$ Total sum $= 27$ * (d) $5 + 5 = 10 \implies$ Total sum $= 27$ *Conclusion:* Every single option sums to 10, meaning every option satisfies the rule of 9. We must rely entirely on the rule of 11. * **Step 2: Check the Rule of 11.** Group the alternating positions (from right to left): * Odd positions: $b + 3 + 2 + 2 + 3 = 10 + b$ * Even positions: $a + 1 + 2 + 4 = 7 + a$ * Difference: $(10 + b) - (7 + a) = 3 + b - a$ For the number to be divisible by 11, the expression $(3 + b - a)$ must equal $0$ or $11$. * If $3 + b - a = 0 \implies a - b = 3$ * If $3 + b - a = 11 \implies b - a = 8$ * **Step 3: Test the options against the derived condition ($b - a = 8$ or $a - b = 3$).** * *(a) 1, 9:* Here $a=1, b=9$. Check: $b - a = 9 - 1 = 8$. (Matches condition perfectly!) * *(b) 3, 7:* Here $a=3, b=7$. Check: $b - a = 7 - 3 = 4$. (Fails) * *(c) 4, 6:* Here $a=4, b=6$. Check: $b - a = 6 - 4 = 2$. (Fails) * *(d) 5, 5:* Here $a=5, b=5$. Check: $b - a = 0$. (Fails) ## Exam Strategy & Shortcut **Look Before You Leap:** The fastest students don't blindly calculate everything. A quick glance at the options reveals that the sum of the digits in every single pair is identical ($1+9=10$, $3+7=10$, etc.). This immediately tells you that checking the rule of 9 is a complete waste of time because the question setter designed it that way. You can skip Step 1 entirely and go straight to formulating the difference equation for the rule of 11, cutting your solving time in half. ## Common Pitfall A very common error is reversing the order of the variables. Because the prompt asks for `**` at the end of the number, the first number in the option corresponds to the tens place ($a$), and the second corresponds to the units place ($b$). Setting up the alternating sum backwards will yield an incorrect difference equation and lead you to confidently select the wrong pair. ## Final Answer **Therefore, the correct answer is 1, 9.**