# Cube Root solution

• May 28th 2009, 09:04 PM
mythili666
Cube 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 PM
TheEmptySet
Quote:

Originally Posted by mythili666
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

Here is how you would do the first one

Step one prime factor the number.

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

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

to be a perfect cube each exponent must be divisable by 3. so the only factor missing is 3.
• June 20th 2009, 06:17 AM
s_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.
• June 29th 2009, 03:10 AM
pfarnall
Quote:

Originally Posted by s_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.

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.