Question 624173
Don't post so many problems in a single post. You will probably never get answers to them all. I will do one and hope that it helps you figure out how to do some or all of the others.<br>
Here's a procedure that will help with #2 and #4 and may help with the others:<ol><li>Perform prime factorization on the number.</li><li>Pair up matching prime factors as much as possible.</li><li>The number you are looking for is the product of all the unpaired prime factors.</li></ol>Lets see how this works on #2:
1. Prime factorization.
294 = 2*147 = 2*3*49 = 2*3*7*7
2. Pair up the prime factors.
Only is only one pair of matching factors:
294 = 2*3*(7*7)
3. The number we're looking for is the product of the unpaired factors: 2*3 or 6.<br>
Hint for #3: As it turns out, the number you divide by to get a perfect square is also the number you multiply by to get a perfect square!<br>
Hint for #5: For perfect cubes you group three matching factors, not just two.