Numbers have some pretty interesting properties. Firstly, lets start with factors. The factors of a number are all of the numbers that evenly divide it. For instance, the factors of 6 (in pairs) are 1 and 6 and 2 and 3. Usually, we list them in numerical order: {1, 2, 3, 6}. The factors of 12 are 1 and 12, 2 and 6, 3 and 4. In order: {1, 2, 3, 4, 6, 12}. Try finding the factors of a few other numbers.
One of the things you should notice is that you don't have to test numbers beyond the square root of the original number, because factors come in pairs, and those pairs slowly get closer and closer together. Looking at the factor pairs for 12 as listed above, we should notice that the middle pair, 3 and 4, tells us that the square root of 12 is somewhere between 3 and 4, since there are no other natural numbers between those two.
Ok, now that we know how to find the factors of a number, let's do some exploring with them. If we add up all of the factors of a number (except for the number itself), one of three things can happen. Either the sum is bigger than the original number, it's smaller than the original number, or it's equal to the original number. The rarest occurrence is having the sum equal the original number. In this case, the number is called a "perfect number". If the sum is bigger than the original number, then the number is called "abundant". If the sum is smaller than the original number, then the number is "deficient".
Some numbers have only 1 and themselves as factors, such as 17 or 31. These numbers are called "prime" and are special. It's easy to find small numbers that are prime, but bigger prime numbers can be a problem. In my next post, I'll talk about a method for finding prime numbers as ancient as numbers themselves. I
n the meantime, have fun finding abundant, deficient and perfect numbers!
2 comments:
Do you add the 1? In that case, 6 is a perfect number?
Yes, you add in the 1. 1 is a factor of every number and is therefore included in every factor list you will ever create. Six is perfect. The next perfect number is 28.
Incidentally, all prime numbers are deficient, since their only factor (other than themselves) is 1. And 1 is always less than the prime number in question.
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