To determine the digits indicated by and \[tex]$ in the number 3422213 \$[/tex] that make the number divisible by 99, we need to consider the conditions for divisibility by both 9 and 11, as 99 is the product of 9 and 11.
Condition for Divisibility by 9:
1. The sum of all the digits must be divisible by 9.
Condition for Divisibility by 11:
1. The alternating sum of the digits (i.e., the sum where we alternately add and subtract the digits) must be divisible by 11.
Step-by-Step Solution:
1. Calculate the sum of the known digits (3, 4, 2, 2, 2, 1, 3) without the positions represented by * and \[tex]$:
\[
3 + 4 + 2 + 2 + 2 + 1 + 3 = 17
\]
Let the digits represented by * and \$[/tex] be [tex]\( x \)[/tex] and [tex]\( y \)[/tex] respectively.
The total sum of the digits including [tex]\( x \)[/tex] and [tex]\( y \)[/tex] will be:
[tex]\[
17 + x + y
\][/tex]
For the number to be divisible by 9, [tex]\( 17 + x + y \)[/tex] must also be divisible by 9.
2. Next, calculate the alternating sum of the digits (without * and \[tex]$):
\[
3 - 4 + 2 - 2 + 2 - 1 + 3 = 3
\]
Including \( x \) and \( y \), the alternating sum will be:
\[
3 + x - y
\]
For the number to be divisible by 11, \( 3 + x - y \) must also be divisible by 11.
Given Choices:
- A) 1, 9
- B) 3, 7
- C) 4, 6
- D) 5, 5
Check each pair:
- For \((x, y) = (1, 9)\):
- Sum: \( 17 + 1 + 9 = 27 \)
- Alternating Sum: \( 3 + 1 - 9 = -5 \)
Neither 27 is divisible by 9, nor -5 is divisible by 11.
- For \((x, y) = (3, 7)\):
- Sum: \( 17 + 3 + 7 = 27 \)
- Alternating Sum: \( 3 + 3 - 7 = -1 \)
Neither 27 is divisible by 9, nor -1 is divisible by 11.
- For \((x, y) = (4, 6)\):
- Sum: \( 17 + 4 + 6 = 27 \)
- Alternating Sum: \( 3 + 4 - 6 = 1 \)
Neither 27 is divisible by 9, nor 1 is divisible by 11.
- For \((x, y) = (5, 5)\):
- Sum: \( 17 + 5 + 5 = 27 \)
- Alternating Sum: \( 3 + 5 - 5 = 3 \)
Neither 27 is divisible by 9, nor 3 is divisible by 11.
Thus, none of the given choices \( (1, 9), (3, 7), (4, 6), (5, 5) \) make the number 3422213 * \$[/tex] divisible by 99.
Therefore, the correct solution to the problem is that there is no pair among the provided options that satisfies both divisibility conditions. Hence, the answer is none of the above.
