A positive integer n is a perfect number provided that the sum of all the positive factors of n, including...

1. Translate the problem requirements: We need to find all positive factors of 28, then calculate the sum of the reciprocals (1 divided by each factor) of all these factors.
2. Verify that 28 is indeed a perfect number: Check that the sum of all factors of 28 equals \(2 \times 28 = 56\), confirming our starting point.
3. Identify all positive factors of 28: Use prime factorization to systematically find every number that divides 28 evenly.
4. Apply the reciprocal relationship for factors: Use the key insight that if d is a factor of n, then \(\frac{n}{d}\) is also a factor, and these reciprocals have a special relationship that simplifies our calculation.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we need to do in everyday language. We have the number 28, and we need to find all the numbers that divide evenly into 28 (these are called factors). Then, for each factor, we need to calculate "1 divided by that factor" (this is called the reciprocal). Finally, we add up all these reciprocals.

For example, if 4 is a factor of 28, then the reciprocal of 4 is \(\frac{1}{4}\). We need to find the sum of all such reciprocals.

Process Skill: TRANSLATE - Converting the problem language into clear mathematical steps

2. Verify that 28 is indeed a perfect number

Before we proceed, let's confirm that 28 is actually a perfect number. A perfect number means that when we add up all its factors (including 1 and the number itself), we get exactly twice the number.

Let's find all factors of 28 first. Since \(28 = 4 \times 7 = 2^2 \times 7\), the factors are:

• 1 (divides every number)
• 2 (since \(28 \div 2 = 14\))
• 4 (since \(28 \div 4 = 7\))
• 7 (since \(28 \div 7 = 4\))
• 14 (since \(28 \div 14 = 2\))
• 28 (the number itself)

Sum of factors: \(1 + 2 + 4 + 7 + 14 + 28 = 56\)
Twice the number: \(2 \times 28 = 56\) ✓

Yes, 28 is indeed a perfect number!

3. Identify all positive factors of 28

We already found all the factors in the previous step, but let's organize them clearly:

Using prime factorization: \(28 = 2^2 \times 7^1\)

All factors are formed by taking different powers of the prime factors:

• \(2^0 \times 7^0 = 1\)
• \(2^1 \times 7^0 = 2\)
• \(2^2 \times 7^0 = 4\)
• \(2^0 \times 7^1 = 7\)
• \(2^1 \times 7^1 = 14\)
• \(2^2 \times 7^1 = 28\)

So our complete list of factors is: 1, 2, 4, 7, 14, 28

4. Apply the reciprocal relationship for factors

Now here's the key insight that makes this problem elegant. When we look at the factors of 28, notice something special:

• If 1 is a factor, then \(\frac{28}{1} = 28\) is also a factor
• If 2 is a factor, then \(\frac{28}{2} = 14\) is also a factor
• If 4 is a factor, then \(\frac{28}{4} = 7\) is also a factor

This means our factors come in pairs that multiply to 28!

Now, when we find the reciprocals:

• \(\frac{1}{1} + \frac{1}{28}\)
• \(\frac{1}{2} + \frac{1}{14}\)
• \(\frac{1}{4} + \frac{1}{7}\)

Let's use the fact that for each pair of factors d and \(\frac{28}{d}\):
\(\frac{1}{d} + \frac{1}{\frac{28}{d}} = \frac{1}{d} + \frac{d}{28} = \frac{28 + d^2}{28d}\)

But there's an even simpler way! Let's just calculate directly:

Sum of reciprocals = \(\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{7} + \frac{1}{14} + \frac{1}{28}\)

To add these fractions, let's use 28 as our common denominator:
\(= \frac{28}{28} + \frac{14}{28} + \frac{7}{28} + \frac{4}{28} + \frac{2}{28} + \frac{1}{28}\)
\(= \frac{28 + 14 + 7 + 4 + 2 + 1}{28}\)
\(= \frac{56}{28}\)
\(= 2\)

Process Skill: SIMPLIFY - Using common denominators to make fraction arithmetic manageable

Final Answer

The sum of the reciprocals of all positive factors of 28 is 2.

Notice the beautiful relationship: for the perfect number 28, the sum of all its factors is 56, and the sum of the reciprocals of all its factors is \(\frac{56}{28} = 2\).

The answer is C) 2.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the definition of perfect number

Students might confuse the definition and think that a perfect number is one where the sum of proper divisors (excluding the number itself) equals the number, rather than understanding that the sum of ALL positive factors (including 1 and the number itself) equals 2n. This could lead them to verify 28 incorrectly or doubt whether 28 is actually a perfect number.

2. Confusion about what "positive factors" includes

Some students might forget to include either 1 or 28 itself when listing the positive factors, thinking that factors only refer to numbers between 1 and the given number (exclusive). This would result in an incomplete list of factors and an incorrect final answer.

3. Not recognizing the reciprocal calculation requirement

Students might misread the question and think they need to find the sum of factors (which is 56) rather than the sum of reciprocals of factors (\(\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{7} + \frac{1}{14} + \frac{1}{28}\)).

Errors while executing the approach

1. Arithmetic errors when finding factors

When using prime factorization (\(28 = 2^2 \times 7\)), students might miss some factors or include incorrect numbers. For example, they might forget that \(14 = 2^1 \times 7^1\) is a factor, or incorrectly include numbers like 8 or 21 that don't actually divide 28.

2. Fraction arithmetic mistakes

When adding the reciprocals \(\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{7} + \frac{1}{14} + \frac{1}{28}\), students often struggle with finding the common denominator or make computational errors. They might use an incorrect common denominator (like 84 instead of 28) or make mistakes when converting fractions.

3. Calculation errors when converting to common denominator

Even with the correct approach of using 28 as the common denominator, students might incorrectly convert fractions. For example, they might write \(\frac{1}{2}\) as \(\frac{13}{28}\) instead of \(\frac{14}{28}\), or make similar conversion errors that lead to an incorrect numerator sum.

Errors while selecting the answer

No likely faltering points - once students correctly calculate the sum as \(\frac{56}{28}\), the simplification to 2 is straightforward, and 2 directly matches answer choice C.