Posts tagged ‘abundant’
A Perfect State
A perfect number is a positive integer that is equal to the sum of its proper positive factors.
Perfect numbers, like perfect individuals, are very rare.
– Rene Descartes
They are very rare, indeed — there are only five perfect numbers less than 1,000,000,000. Because 6 and 28 are two of them, you might say that today is a perfect day.
Perfect Square: A nerd who never makes mistakes.
I recently coined the term perfect state to refer to a state for which the number of letters in its name is equal to the number of letters in the name of the state capital.
To pass the time on a recent car trip, I asked my sons to see how many perfect states they could find. During their search, they identified many states that were not perfect, and they giggled gleefully when I referred to them as abundant (more letters in the capital than in the state) and deficient (fewer letters in the capital than in the state).
They were extremely excited to learn that our home state, Virginia, is a perfect state. This did not surprise me — with beaches to the east, mountains in the west, urban living in the north, rural country in the south, and a whole lot of wine country in between, I’ve often argued that Virginia is the perfect state. (In their song Old Dominion, local band Eddie from Ohio describes Virginia as “just southeast of heaven to the surf and the hills.” Yeah, that’s about right.)
Anyway, my sons were able to find Virginia and seven other perfect states without the help of a map. Can you?
Need some help? Check out this map with a colorcoded solution. The deficient states are white, the abundant states are dark blue, and the perfect states are light blue.
Fun Factor
Two numbers were having a conversation about their social lives.
28: Did you hear that 284 broke up with 220?
6: I’m not surprised. He’s far from perfect. But at least their breakup was amicable.
28: Yeah, well, I heard she started seeing 12.
6: Really? He doesn’t have abundant charm. Don’t you think 10 would be a better match for her?
28: I don’t know. He seems so solitary!
Speaking of factors, I learned a neat trick this weekend for finding the sum of the factors of a number. Before I share that, consider the method for determining how many factors a number has. Take the number 12, for instance. The prime factorization of 12 is:
12 = 2^{2} × 3
The following array can be used to generate all of the factors of 12:

2^{0}  2^{1}  2^{2} 
3^{0}  1  2  4 
3^{1}  3  6  12 
It’s obvious from the array that there are six factors. But the trick is to notice that each factor in the array is made from a power of 2 times a power of 3 — that is, each factor is equal to 2^{m} × 3^{n}, where 0 ≤ m ≤ 2 and 0 ≤ n ≤ 1. Since there are three possible values of m and two possible values for n, then there are 3 × 2 = 6 factors of 12.
In general, if the prime factorization of the number takes the form a^{p} × b^{q} × c^{r}, then the number of factors is (p + 1)(q + 1)(r + 1) for exponents p, q, and r. (The process could obviously be extended if there are more than three prime factors.)
But look at the array again. The sum of all factors of 12 is equal to sum of all products that occur within the array. However, there is an easy way to find that sum, by taking advantage of the distributive property. The sum of the powers of 2 along the top is 2^{0} + 2^{1} + 2^{2} = 7, and the sum of the powers of 3 along the left side is 3^{0} + 3^{1} = 4. Consequently, the sum of all factors of 12 is equal to:
(2^{0} + 2^{1} + 2^{2})(3^{0} + 3^{1}) = 7 × 4 = 28
In general, if the prime factorization of a number is a^{p} × b^{q} × c^{r}, then the sum of the factors is:
(a^{0} + a^{1} + … + a^{p})(b^{0} + b^{1} + … + b^{q})(c^{0} + c^{1} + … + c^{r})
And again, this could be extended if the number had more than three prime factors.
Cool, huh?