- Translate the problem requirements: We need to find all 1-digit values of x where \(3305 \times \mathrm{x}\) gives a 5-digit number with units digit 5, then find all possible hundreds digits y in these products. The sets A and B contain these possible values.
- Apply the 5-digit constraint: Determine which values of x make \(3305 \times \mathrm{x}\) fall between 10,000 and 99,999 to ensure it's exactly 5 digits.
- Use units digit multiplication pattern: Since 3305 ends in 5, find which single digits x make the units digit of \(3305 \times \mathrm{x}\) equal to 5.
- Calculate actual products and extract hundreds digits: For valid x values, compute \(3305 \times \mathrm{x}\) and identify the hundreds digit in each result to form set B.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what this problem is asking us to find in everyday terms:
- We have the number 3305 and we're multiplying it by some single digit (1, 2, 3, 4, 5, 6, 7, 8, or 9)
- The result must be a 5-digit number (between 10,000 and 99,999)
- The last digit (units digit) of our result must be 5
- We need to find all possible single digits x that work (this becomes set A)
- For each valid x, we look at the hundreds digit of the result (this becomes set B)
Think of it like this: if \(3305 \times \mathrm{x} = \mathrm{abcde}\) (where abcde is a 5-digit number), then e = 5, and we want to collect all possible values of c (the hundreds digit).
Process Skill: TRANSLATE
2. Apply the 5-digit constraint
For the product to be exactly 5 digits, it must be at least 10,000 but less than 100,000.
Let's find the range of x values:
- Smallest 5-digit number: \(10,000 \div 3305 \approx 3.02\)
- Largest 5-digit number: \(99,999 \div 3305 \approx 30.25\)
Since x must be a single digit (1-9), we need \(\mathrm{x} \geq 4\) to get at least a 5-digit result. Let's check:
- \(3305 \times 3 = 9,915\) (only 4 digits - too small)
- \(3305 \times 4 = 13,220\) (5 digits - good!)
So x must be at least 4, and since x is a single digit, x can be 4, 5, 6, 7, 8, or 9.
Process Skill: APPLY CONSTRAINTS
3. Use units digit multiplication pattern
Now we need the units digit of the product to be 5. Since 3305 ends in 5, we need to find which single digits, when multiplied by 5, give a units digit of 5.
Let's check the multiplication table for 5:
- \(1 \times 5 = 5\) (units digit is 5 ✓)
- \(2 \times 5 = 10\) (units digit is 0 ✗)
- \(3 \times 5 = 15\) (units digit is 5 ✓)
- \(4 \times 5 = 20\) (units digit is 0 ✗)
- \(5 \times 5 = 25\) (units digit is 5 ✓)
- \(6 \times 5 = 30\) (units digit is 0 ✗)
- \(7 \times 5 = 35\) (units digit is 5 ✓)
- \(8 \times 5 = 40\) (units digit is 0 ✗)
- \(9 \times 5 = 45\) (units digit is 5 ✓)
So the units digit is 5 when x is odd: 1, 3, 5, 7, or 9.
Combining with our 5-digit constraint (\(\mathrm{x} \geq 4\)), we get: x can be 5, 7, or 9.
Wait, let's double-check x = 3: \(3305 \times 3 = 9,915\) (only 4 digits), so x = 3 doesn't work.
Therefore, set A = {5, 7, 9}.
Process Skill: CONSIDER ALL CASES
4. Calculate actual products and extract hundreds digits
Now let's calculate the actual products for x = 5, 7, and 9, and find the hundreds digit in each:
- \(3305 \times 5 = 16,525\)
Breaking this down: 1-6-5-2-5 (ten thousands, thousands, hundreds, tens, units)
The hundreds digit is 5
- \(3305 \times 7 = 23,135\)
Breaking this down: 2-3-1-3-5
The hundreds digit is 1
- \(3305 \times 9 = 29,745\)
Breaking this down: 2-9-7-4-5
The hundreds digit is 7
Therefore, set B = {1, 5, 7}.
5. Final Answer
We found:
- Set A (possible values of x) = {5, 7, 9}
- Set B (possible values of y, the hundreds digit) = {1, 5, 7}
Looking at the answer choices, this matches option (D): {5, 7, 9}----------{1, 5, 7}
Answer: D
Common Faltering Points
Errors while devising the approach
- Missing the 5-digit constraint: Students often focus only on finding values of x where the units digit is 5, but forget that the product must specifically be a 5-digit number. This leads them to include x = 1 and x = 3 in set A, even though \(3305 \times 1 = 3,305\) (4 digits) and \(3305 \times 3 = 9,915\) (4 digits) don't meet the requirement.
- Misunderstanding what the hundreds digit means: Students may confuse which digit position represents the hundreds place in a 5-digit number. For a number like 23,135, they might incorrectly identify 3 (thousands digit) or 3 (tens digit) as the hundreds digit instead of the correct digit 1.
- Overlooking the constraint that x must be a 1-digit integer: Some students might consider x = 0 as a possibility, not realizing that multiplying by 0 would give a product of 0, which is not a 5-digit number, or they might not restrict themselves to single digits 1-9.
Errors while executing the approach
- Arithmetic errors in multiplication: Students may make calculation mistakes when computing products like \(3305 \times 7 = 23,135\) or \(3305 \times 9 = 29,745\). Even small errors here will lead to wrong hundreds digits for set B.
- Incorrect boundary calculations: When determining the range for x to ensure a 5-digit result, students might make errors in division (like \(10,000 \div 3305\)) or incorrectly round the results, leading them to include or exclude wrong values of x.
- Units digit pattern errors: Students might incorrectly determine which values of x give a units digit of 5 when multiplied by 5, possibly confusing the pattern or making errors in the multiplication table for 5.
Errors while selecting the answer
- Mixing up sets A and B: Students might correctly find the values {5, 7, 9} and {1, 5, 7} but accidentally reverse which set corresponds to A (possible x values) versus B (possible hundreds digits), leading them to select an incorrect answer choice.
- Including invalid x values in the final answer: Even after discovering that x = 1 and x = 3 don't produce 5-digit numbers, students might still include these in their final set A, selecting answer choice (B) instead of the correct (D).
