October 13, 2017
Happy Friday the 13th! Did you know that fear of the number thirteen is called triskaidekaphobia? I heard that on the news when I was in 5th grade, and I never forgot it. However, I don’t think I knew how to spell it.
Speaking of 13, 13 is a prime number. In the fifth grade math classroom, we’ve been investigating primes, composites, odds, and evens. We’ve been thinking about what happens to the number of factors when we double numbers.
It all started when we were looking at factors of 12, 24, 48, and 96. We noticed that each time we doubled, we gained 2 new factors. We then questioned whether or not that always happens. With prime numbers (such as 13), it does. For example, the factors of 13 are 1 and 13. When we double 13 to get 26, we get 2 additional factors (2 and 26). It seemed there was a phenomenon there that always occurred.
Then, we continued to explore, and we found some counter-examples. When we doubled 2, we only gained 1 factor. When we doubled 21, we gained 4 factors. What was the reason for this? Ultimately, we found out that this gaining 2 factors idea works for all even numbers, except if there’s a square number involved. Four is square, so it’s factors are 1, 2, and 4. Since it has an odd number of factors (since 2 is multiplied by itself), it didn’t gain a whole factor pair.
As for 21 (as with all odd numbers), we realized that the number of factors doubles when you double the number. Since all the factors of an odd number are odd, when you double it, all of the factors also get doubled, giving us an even factor for every odd. So 21’s factors are 1, 3, 7, 21. Then when we double to get 42, we have the original 4 odd factors and 4 new even factors: 1, 2, 3, 6, 7, 14, 21, 42. So, while it is true that prime numbers gain 2 factors , as in our original conjecture, it is because they only start with 2 factors, so when we double we get 4.
Have a great weekend. Don’t be afraid of 13!