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Factor 756 into prime factors. Then add additional prime factors, such that each prime factor occurs a number of times that is a multiple of 3. The product of the additional prime factors is "k".

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Q: Find the smallest integer k such that 756k is a perfect cube?
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What is the second largest whole number which is a perfect square a perfect cube and a perfect fifth power?

The second LARGEST? Is that correct? I think the second SMALLEST is a much more sensible question. How could you possibly know which is the LARGEST, much less the second largest? The SMALLEST is of course 1. Since 1^2 = 1, 1^3 = 1 and 1^5 = 1. The second SMALLEST I could find is 1073741824. I didn't try all possible numbers, but that was the second smallest I could find. 1024^2 = 1073741824, 32768^3 = 1073741824 and 64^5 = 1073741824. My initial gut was 64, but it isn't a perfect 5th, 2^5 = 32 NOT 64. Just try a couple 5th powers and see which are factorable (into a perfect square and a perfect cube). If you have a graphing calculator (or a computer) you can use the 3rd root and square root functions to do the math for you. But 64^5 was the smallest I could find (other than 1). Other numbers like 12^5, 24^5 and 32^5 did not work-out but 64 did. Hope this helps!


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