Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

## #1 2012-11-17 11:58:27

zehao1000
Member

Offline

### Question on squares

I just want to to know how to solve this problem:

Maya lists all the positive divisors of 2012^2. She then randomly selects 2 of them divisiors. Let p be the probability that exactly one of the 2 divisors is a square. p can be expressed as m/n where m and n are relatively primes numbers. Find m+n.

I solved to a part that the p(of a square) 16/81 and p(of non-sqaure) 65/81. What to do next???????

## #2 2012-11-17 12:21:50

bobbym

Online

### Re: Question on squares

Hi;

She then randomly selects 2 of them divisiors.

With replacement or without?

In mathematics, you don't understand things. You just get used to them.
Some cause happiness wherever they go; others, whenever they go.
If you can not overcome with talent...overcome with effort.

## #3 2012-11-17 12:36:17

zehao1000
Member

Offline

### Re: Question on squares

The 2 divisors do not have replacement, and the 2 divisors are distinct, thanks.

## #4 2012-11-17 12:45:03

bobbym

Online

### Re: Question on squares

Hi;

Does your list of divisors look like this?

In mathematics, you don't understand things. You just get used to them.
Some cause happiness wherever they go; others, whenever they go.
If you can not overcome with talent...overcome with effort.

## #5 2012-11-17 12:50:55

zehao1000
Member

Offline

### Re: Question on squares

Actually, the number of divisors that 2012^2 has is (2+1)^4, the problem is to find the sum of the numerator+denomerator of the probability of the chance that only ONE of the two divisors is a perfect square, the number of divisors that 2012 has is (1+1)^4 and this is also the number of perfect square divisors 2012^2 has, I'm very bad at probability and can't figure what is the probability of only one of the divisors being a perfect square.

## #6 2012-11-17 12:55:19

bobbym

Online

### Re: Question on squares

Hi;

Can you post the divisors of 2012 ^2 please?

I am getting 16 and proper divisors only 15.

In mathematics, you don't understand things. You just get used to them.
Some cause happiness wherever they go; others, whenever they go.
If you can not overcome with talent...overcome with effort.

## #7 2012-11-17 13:13:04

zehao1000
Member

Offline

### Re: Question on squares

I can't list them all, let's see.....: 1,2,4,16,253009,1012036, 2024072,4048144, these are only the half of the perfect square factors of 2012^2, there are 8 more of them, how to solve the problem goes like this: (2+1)^4 is the number of factors that 2012^2 has and (1+1)^4 is the number of factors that 2012^2 has that are perfect squares, so the probability is [2*2^4*(3^4-2^4)]/[3^4(3^4-1)=26/81 so the answer is 26+81=107, which is m+n, this looks more like a bunch of random numbers placed together and I personally think that knowing the answer with knowing how to do it is useless, so I just need an explanation of this, you are only given 12 minutes to do this, I don't think listing out all the positive divisors of 2012^2 is a efficient idea!

## #8 2012-11-17 13:15:47

bobbym

Online

### Re: Question on squares

I am not claiming it is an efficient idea but I have to know the sample space before we can get the probability. I am getting only 15 divisors of 2012^2 not 81. That is why I am asking for your list.

In mathematics, you don't understand things. You just get used to them.
Some cause happiness wherever they go; others, whenever they go.
If you can not overcome with talent...overcome with effort.

## #9 2012-11-17 13:31:55

zehao1000
Member

Offline

### Re: Question on squares

It does look weird that 2012^2 has that many factors, but 2012 has a prime factorization of 2*3*5*67, so 2012^2 has a prime factorization of 2^2*3^2*5^2*67^2 and that leads to the fact that (2+1)^4 is the number of factors that 2012^2 has, though only 16 of them are perfect squares, so the question is simplified to what is the probability of getting only 1 perfect square out of 2 if the probability of getting a square is 16/81. I also doubt that 2012^2 has 81 factors, but that's what the formula calculates, but I think I get the question now, but anyways, sorry to bother you with all these random things I said above, the only reason why I want to have the answer to this problem is because it fascinates me and is a really good probability problem for me.

## #10 2012-11-17 13:37:51

bobbym

Online

### Re: Question on squares

Hi;

That is not correct 2012^2 only has 15 factors not 81. The list I gave in post #4 are the only positive factors.

so 2012^2 has a prime factorization of 2^2*3^2*5^2*67^2

You have factored 2012^2 incorrectly.

In mathematics, you don't understand things. You just get used to them.
Some cause happiness wherever they go; others, whenever they go.
If you can not overcome with talent...overcome with effort.

## #11 2012-11-17 13:46:07

zehao1000
Member

Offline

### Re: Question on squares

OH-NO, sorry! I got the whole problem wrong! It's suppose be 2010, not 2012 and 2010^2 has 81 factors, I can't believe that just a difference of 2 can make such a huge difference, now I hope that this problem is much more clear! I actually though that 5 was a factor of 2012!

## #12 2012-11-17 13:47:22

bobbym

Online

### Re: Question on squares

Hi;

Yes, 2010^2 has 81 factors.

In mathematics, you don't understand things. You just get used to them.
Some cause happiness wherever they go; others, whenever they go.
If you can not overcome with talent...overcome with effort.

## #13 2012-11-17 13:52:54

zehao1000
Member

Offline

### Re: Question on squares

Because this problem was in a 2010 contest so though I still thought that it was 2012, but now the problem seems to make much more sense.

## #14 2012-11-17 13:56:25

bobbym

Online

### Re: Question on squares

I am getting 26 / 81 as the probability one of the numbers is a square.

In mathematics, you don't understand things. You just get used to them.
Some cause happiness wherever they go; others, whenever they go.
If you can not overcome with talent...overcome with effort.

## #15 2012-11-17 13:58:49

zehao1000
Member

Offline

### Re: Question on squares

Yes, that is correct, I'm just trying to understand the solution by myself, so if 26/81 is the probability, then m+n is equal to 26+81 which is 107, which is the final answer, but the solution is confusing, since it's all probability.

## #16 2012-11-17 14:02:58

bobbym

Online

### Re: Question on squares

Hi;

In mathematics, you don't understand things. You just get used to them.
Some cause happiness wherever they go; others, whenever they go.
If you can not overcome with talent...overcome with effort.

## #17 2012-11-17 14:05:44

zehao1000
Member

Offline

### Re: Question on squares

Oh, thanks! Now I get the whole thing.

## #18 2012-11-17 14:14:59

bobbym

Online

### Re: Question on squares

Hi;

You are welcome!

In mathematics, you don't understand things. You just get used to them.
Some cause happiness wherever they go; others, whenever they go.
If you can not overcome with talent...overcome with effort.