How many positive integers less than 20 are either a multiple of 2, an odd multiple of 9, or the...

1. Translate the problem requirements: We need to count positive integers less than 20 that fall into ANY of three categories: (1) multiples of 2, (2) odd multiples of 9, or (3) numbers that can be expressed as (positive multiple of 2) + (positive multiple of 9)
2. Identify and list each category systematically: Find all numbers in each of the three categories within our range of 1-19
3. Combine categories using union principle: Since we want numbers that satisfy ANY condition, we take the union of all three sets, being careful not to double-count
4. Count the total unique numbers: Determine how many distinct positive integers less than 20 satisfy at least one of our three conditions

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we're looking for in plain English. We need to count positive integers less than 20 (so numbers from 1 to 19) that satisfy ANY of these three conditions:

Category 1: Multiples of 2 (even numbers)
Category 2: Odd multiples of 9
Category 3: Numbers that can be written as (some positive multiple of 2) + (some positive multiple of 9)

The key word here is "either" - this means a number qualifies if it fits into ANY of these three categories. We don't need all three conditions; just one is enough.

Process Skill: TRANSLATE - Converting the complex problem statement into clear, manageable categories

2. Identify and list each category systematically

Now let's find all the numbers in each category within our range of 1 to 19.

Category 1: Multiples of 2 (even numbers from 1 to 19)
These are simply the even numbers: 2, 4, 6, 8, 10, 12, 14, 16, 18

Category 2: Odd multiples of 9
Multiples of 9 less than 20 are: 9, 18
Of these, which are odd multiples? We need 9 × (odd number):
• \(9 \times 1 = 9\) ✓ (odd multiple)
• \(9 \times 2 = 18\) ✗ (even multiple)
So Category 2 contains: 9

Category 3: Sum of (positive multiple of 2) + (positive multiple of 9)
Positive multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, ...
Positive multiples of 9: 9, 18, 27, ...
We need sums less than 20:
• \(2 + 9 = 11\) ✓
• \(4 + 9 = 13\) ✓
• \(6 + 9 = 15\) ✓
• \(8 + 9 = 17\) ✓
• \(10 + 9 = 19\) ✓
• \(12 + 9 = 21\) ✗ (too big)
So Category 3 contains: 11, 13, 15, 17, 19

3. Combine categories using union principle

Since we want numbers that satisfy ANY condition, we take the union of all three sets. Let's list all numbers and avoid double-counting:

From Category 1: 2, 4, 6, 8, 10, 12, 14, 16, 18
From Category 2: 9
From Category 3: 11, 13, 15, 17, 19

Combining all unique numbers: 2, 4, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19

Process Skill: CONSIDER ALL CASES - Making sure we account for all numbers without double-counting

4. Count the total unique numbers

Let's verify our list by checking which numbers from 1 to 19 are NOT included:

Numbers from 1 to 19: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19

Our qualifying numbers: 2, 4, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19

Numbers NOT included: 1, 3, 5, 7

Let's double-check why these four don't qualify:
• 1: Not even, not an odd multiple of 9, and can't be written as (multiple of 2) + (multiple of 9) since the smallest such sum is \(2 + 9 = 11\)
• 3: Not even, not a multiple of 9, and \(3 = 2a + 9b\) has no positive integer solutions
• 5: Not even, not a multiple of 9, and \(5 = 2a + 9b\) has no positive integer solutions
• 7: Not even, not a multiple of 9, and \(7 = 2a + 9b\) has no positive integer solutions

Counting our qualifying numbers: 15 numbers total.

Final Answer

We found 15 positive integers less than 20 that satisfy at least one of the three conditions. The answer is E.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "odd multiple of 9" as "multiples of 9 that are odd numbers"
Students often confuse this phrase and think it means "take all multiples of 9 and pick the odd ones" (which would give them 9 only). The correct interpretation is "9 multiplied by an odd number" (\(9 \times 1 = 9\), \(9 \times 3 = 27\), etc.). This distinction is crucial but easily missed.

2. Treating "either...or" as exclusive rather than inclusive
The word "either" can confuse students into thinking the categories are mutually exclusive - that a number can only belong to one category. In reality, this is an inclusive "or" where numbers can satisfy multiple conditions simultaneously, and we want the union of all qualifying numbers.

3. Overlooking the constraint that multiples must be positive
The problem specifies "positive multiple of 2" and "positive multiple of 9" for Category 3, but students might miss this constraint and consider zero as a valid multiple, potentially affecting their calculations for sums.

Errors while executing the approach

1. Incomplete enumeration of Category 3 sums
When finding numbers that can be expressed as (multiple of 2) + (multiple of 9), students often stop too early or miss systematic checking. They might only try \(2 + 9 = 11\) and forget to check \(4 + 9 = 13\), \(6 + 9 = 15\), etc., leading to an incomplete list.

2. Double-counting numbers that appear in multiple categories
Since some numbers satisfy multiple conditions (like 18 being both even and potentially expressible as a sum), students might count them multiple times instead of taking the union. This leads to overcounting and an incorrect final answer.

3. Arithmetic errors when checking Category 3 exclusions
When verifying that numbers like 1, 3, 5, 7 cannot be written as \(2a + 9b\) with positive integers a and b, students might make computational errors or not systematically check all possibilities, leading them to incorrectly include or exclude numbers.

Errors while selecting the answer

No likely faltering points - the final counting step is straightforward once the correct set of qualifying numbers is identified.