{"id":780,"date":"2014-02-25T14:43:23","date_gmt":"2014-02-25T06:43:23","guid":{"rendered":"http:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/2014\/02\/perfect-numbers\/"},"modified":"2024-02-16T21:12:10","modified_gmt":"2024-02-16T13:12:10","slug":"perfect-numbers","status":"publish","type":"post","link":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/2014\/02\/perfect-numbers\/","title":{"rendered":"Perfect Numbers"},"content":{"rendered":"

[This is a back-issue of one of this site’s newsletters<\/a>]<\/em><\/p>\n

Every night I brush my son’s teeth. The dentist said we need to keep doing this until he’s at least 8 or 9.<\/p>\n

<\/p>\n

We use a fancy electric toothbrush with a little digital clock. The clock counts down two minutes – thirty seconds for each quarter-mouth. This year, I’ve taken to counting along with my son the total time I’ve spent brushing his teeth for the whole year.<\/p>\n

This will teach him about time. Or something. Hopefully. We’re up to 102 minutes so far. You can tell we’ve missed a day or two.<\/p>\n

As we go along, I’ve started pointing out interesting features of some of the numbers. “Wow, 97 minutes! That’s special! The only way to write 97 as something times something is as 1 x 97!”<\/p>\n

To entertain myself as the toothbrush hums, I play a game with the numbers. Add up all the factors of the number, except the number itself. Then do this again. And again.<\/p>\n

So, 100, having factors 1, 2, 4, 5, 10, 20, 25 and 50, gives 1+2+4+5+10+20+25+50, which is 117.<\/p>\n

117 has factors 1, 3, 9, 13 and 39, which add up to 65.<\/p>\n

65 gives 1+5+13, which is 19, and there the game ends, because 19 is prime. Its only proper factor is 1.<\/p>\n

The sum you’re working out each step is called the “Aliquot sum” of the number.<\/p>\n

Rarely, the game ends on a ‘perfect’ number – such as 6, which equals 1+2+3, or 28, which equals 1+2+4+7+14. The only perfect numbers known are even, and even they are as rare as hen’s teeth. An even perfect number must be a power of two times a prime, and the prime must be one less than a power of two.<\/p>\n

If working out all the factors is too much work, there’s an easier way to get the Aliquot sum if you have the prime factorization. Let’s try it with 100, which is 22<\/sup> x 52<\/sup>. If I add up all the powers of 2 up to 22<\/sup>, that’s 1+2+4 = 7. Then, I’ll add up all the powers of 5 up to 52<\/sup>, that’s 1+5+25 = 31. Multiply 7 x 31, and you get the sum of all<\/em> the factors of 100, including 100 itself. So, to get the Aliquot som, I need to subtract 100 from 7 times 31, giving 217 – 100 = 117.<\/p>\n

Why don’t you try this game some time, say when brushing your teeth? See what happens when you start with 220, or 276. Let me know!<\/p>\n

Yours,
\nMichael Hartley<\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

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