Please find the smalles number by which the following numbers must be multiplied so that the product has a cube root?

1. 3087

2. 33275

Request provide step by step solution.

Thanks

Ram

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- May 28th 2009, 09:04 PMmythili666Cube Root solution
Please find the smalles number by which the following numbers must be multiplied so that the product has a cube root?

1. 3087

2. 33275

Request provide step by step solution.

Thanks

Ram - May 28th 2009, 09:10 PMTheEmptySet
Here is how you would do the first one

Step one prime factor the number.

Notice that $\displaystyle 3087$ is divisable by 9 so we get

$\displaystyle 3087=9\cdot 343=3^2\cdot 7^3$

to be a perfect cube each exponent must be divisable by 3. so the only factor missing is 3. - Jun 20th 2009, 06:17 AMs_ingram
Hi there,

The second answer is 5 because

33275 = 5 * 5 * 11 * 11 * 11

and the cube root will be 55. But I guess this doesn't help because you seem to be looking for a systematic way of finding these factors and I have to admit I did it almost by witchcraft. I saw the 275 at the end, thought 5, went with another 5 and when I got 1331 I kind of went for 11 straight away. The only way I can see to do it is look for the factors and bunch them in 3s. - Jun 29th 2009, 03:10 AMpfarnall
Actually you have provided a systematic method, you just seem to have stumbled upon it. As in the first question you've found the prime factors.

Start with a list of the prime numbers in order : 2,3,5,7,11...

Try dividing the value by these to find the smallest prime which goes into the value with no remainder, so in this case:

33275/2 (clearly this will not be an integer as 33275 is odd)

33275/3 (doesn't work either - use a calculator if in doubt)

33275/5 = 6655

6655/5 = 1331

1331/5 (clearly won't work as it doesn't end with either a 0 or 5)

1331/7 (nope)

1331/11 = 121

121/11 = 11

and finally

11/11 = 1

Then you have the prime factors.

That's about as systematic as it gets I'm afraid.