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 $\displaystyle 3087$ 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.
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.