Factors
Since 2 
In summary, the factors of 6 are 1, 2, 3, and 6. Excluding the number itself, 1, 2 and 3 are 6’s proper factors. Proper factors impart numbers with interesting properties, as we will see in the remainder of this post.
Amicable numbers
In a beautiful novel entitled “The Housekeeper and the Professor” by Japanese author Yoko Ogawa, an elderly professor learns his housekeeper’s birthday and exclaims:
“Your birthday is February twentieth. Two twenty. Can I show you something? This was a prize I won for my thesis on transcendent number theory when I was at college.” He took off his wristwatch and held it up for me to see. … The inscription on the back of the case read President’s Prize No. 284.
… “Take a break from your dishes for a moment and think about these two numbers: 220 and 284. Do they mean anything to you?”
The professor then writes down the proper factors of each of the two numbers:
220: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110
228: 1, 2, 4, 71, 142
He then leads the housekeeper to this discovery:
220: 1+2+4+5+10+11+20+22+44+55+110 = 228
228: 1+2+4+71+142 = 220
“That’s right! The sum of the factors of 220 is 284, and the sum of the factors of 284 is 220. They’re called ‘amicable numbers,’ and they’re extremely rare. … They’re linked to each other by some divine scheme, and how incredible that your birthday and this number on my watch should be just such a pair.” (Ogawa, pp. 16-19)
Note that “amicable” simply means “friendly”.
Mathematicians’ fascination with amicable numbers
It is said that Pythagoras, circa 500 B.C., was asked: “What is a friend?” He replied: “One who is the other I, such as are 220 and 284.” (Amicable Numbers, Prime Glossary).
In 17th century Europe, Fermat and Descartes, of the famous Cartesian plane, found one pair each. Imagine finding that 9,363,584 and 9,437,056 are an amicable pair without access to a calculator, let alone a computer! In the 18th century, Euler, whose acquaintance we have made before, found over 60 pairs. It took till the early 20th century for a sixteen year old Italian, Nicolo Paganini (not the violinist), to find that 1184 and 1210 were also an amicable pair.
When I was still at school, I fancied a geeky girl but she put put me in the friendzone by saying, “You are the 1184 to my 1210!” That was not a true story!
Other number properties imparted by factors
Proper factors can bestow other properties than amicability on positive whole numbers. If you were to add up the proper factors of such numbers, you may stumble on some interesting facts. In what follows, we will explore the sums of the proper factors of the counting numbers between 1 and 12. We will skip 1 and any other number for which 1 is the only proper factor.
The pattern in Figure 1 above continues:
- In most cases, the sum of the proper factors of a positive whole number is less than the number itself. These numbers, like 4, 8, 9 and 10, are termed deficient numbers.
- In some cases, the sum of the proper factors of a number is greater than itself. This is what we saw with the number 12. Such numbers are termed abundant numbers.
- On rare occasions, we find a number whose proper factors add up to the number itself. In our work above, we saw that 6 was such a number. These are known as perfect numbers. 6, 496 and 8128 are examples of perfect numbers.
Over to you
I have deliberately left out a perfect number between 25 and 30. Why don’t you try to identify this number by following the same process presented in Figure 1? Let me know, in the comments, which is this number.
Further reading
For those of you who are more mathematically inclined, this post on the Explaining Science blog, goes deeper into some of the concepts presented above.
Novel: Ogawa, Y. (2010). The Housekeeper and the Professor (S. Snyder, Trans.). Vintage. (Original work published 2003).
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