Please find the smalles number by which the following numbers must be multiplied so that the product has a cube root?
Request provide step by step solution.
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.
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)
1331/11 = 121
121/11 = 11
11/11 = 1
Then you have the prime factors.
That's about as systematic as it gets I'm afraid.