nycgirl212 wrote:
A positive integer n is a perfect number provided that the sum of all the positive factors of n, including 1 and n, is equal to 2n. What is the sum of the reciprocals of all the positive factors of the perfect number 28?
A) 1/4
B) 56/27
C) 2
D) 3
E) 4
Here is a methodical approach to solve this question.
Given:
A positive integer \(N\) is considered a perfect number if the sum of its factors is equal to \(2N\).
Eg: \(6\) is a perfect number since \(1 + 2 + 3 + 6 = 12 = 2*6\)
Given that \(28\) is a perfect number.
Required:
Sum of reciprocals of the factors of \(28\).
Point to keep in mind:
Note that
every factor of number N is always multiplied with
another factor to get N
Eg:
1 *
6 =
6;
2 *
3 =
6;
3 *
2 =
6;
6 *
1=
6Approach:
1. To calculate the sum of reciprocals, take the common denominator as \(N\) itself (\(28\) in this case).
a. For instance, if we consider \(6\) as our example number, the sum of reciprocals of factors is \(\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{6} = \frac{(6 + 3 + 2 + 1)}{6}\)
2. Since all factors of \(28\) are distinct, the numerator of each fraction becomes the corresponding “other factor” of \(N\).
a. Eg: \(\frac{1}{2}\) is same as \(\frac{3}{6}\)
3. So, the numerator of the sum is simply the sum of all factors.
a. In the case of our example number \(6\), you could see that the sum of reciprocals of factors is \(\frac{(6 + 3 + 2 + 1)}{6}\) and the numerator is simply the sum of factors of \(6\) itself.
4. We know the sum of factors of the given number \(28\)
a. It is given that \(28\) is a perfect number and that the sum of factors of a perfect number is twice the number itself.
Working Out:
1. Sum of reciprocals of the factors of 28 \(= S = \frac{(Sum Of Factors of 28)}{28}\)
2. \(S = 2*28/28 = 2\)
Correct Answer:
Option CNote: I haven’t focused on listing down the factors of \(28\) since that has been done in each of the posts above. Please refer to the previous posts to check the calculations. My primary objective in this post is to highlight the underlying logic in the question: the reason that the question clearly states the definition of a perfect number and explicitly tells us that \(28\) is a perfect number.
Point To Note:
Although it is not incorrect to list down the factors of the given number, such an approach is not a scalable approach. For instance, had the question mentioned \(496\) (which is also a perfect number) instead of \(28\), listing down the factors would have been way too cumbersome and some might even begin to deem the question as “unsolvable in 2 minutes”. However, if you're someone who focused on the logic mentioned above, you would immediately identify that even in the case of \(496\) (or any other perfect number), the sum of reciprocals of all its factors is \(2\) again.
There is a reason that the question gives us the definition (and the property) of a perfect number and tells us explicitly that \(28\) is a perfect number. The test maker wants to see if the test taker (you) can identify and use the given property of a perfect number combined with a common observation to arrive at the answer elegantly. (The observation is reiterated as a Takeaway below.)
Takeaway:
For a positive integer \(N\),
the sum of reciprocals of factors is simply \(\frac{(Sum Of Factors Of N)}{N}\)
Footnote: Do you need to
remember this result? Not really. As long as you
understand how this has been arrived at, you’d come up with many such interesting results while solving any good question using a methodical approach.
Hope this helps.
Cheers,
Krishna