# Puzzle: How many factors does 72,000,000 have?

October 21, 2009 pm31 8:53 pm

This puzzle has been posted here before. (although just as backdrop to a teacher mistake). But it is still fun.

1, 2, 3, 4, 5, 6, … and a whole bunch more.

Can you use this with kiddies? How much would you steer them?

And, for that matter, have you answered this one before? If not, give it a try.

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Most of my experienced kids could probably solve this one — we have done problems like this before. I think this kind of problem is great for helping kids understand what it really means for numbers to be “made up of” factors (plus some combinatorics). We also talk about the sum of all the factors, but that’s not something most of them really “get” as they don’t have enough algebra to appreciate what the distributive law does when the formula is expanded.

I get excited by these, so I’m gonna take a stab.

72 million = 72 * 1 million, so since 72 = 2^3 * 3^2 and 10^6 = 1 mil, so the prime factorization is 2^9 times 3^2 times 5^6. I’m interested in a good method for counting them (I haven’t done Combinations/Permutations in quite a while), but for now I’m off to count sheep.

My initial thought was 2^17, but in trying to eliminate duplicates, I think I got to a more appropriate answer (by a more appropriate method) of 210.

Am I right?

Hint for Nick: keep in mind that every factor will have some subset of the prime factors you identified.

Sara, I agree with your answer.

It’s interesting to see how kids attack this. I usually suggest looking at smaller numbers, but sometimes I offer them a few, and sometimes I let them flounder. The floundering takes longer, but I think they get more out of it.

And yes, 210.

Floundering is, IMO, exactly that part of the process in which you do the learning.

I might give this to some of my older kids, unhinted, as a puzzle to attack.