Answer: 0.61
This value is exact without any rounding done to it.
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Step-by-step explanation:
Get some graph paper or open up a spreadsheet. Create a 10 by 10 grid that will hold the numbers 1,2,3,...,98,99,100.
A number is considered square-free when a perfect square is not a factor. Ignore 1 since it is a factor of everything.
Let's look at the multiples of 4:
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104
We stop here since we've gone over 100. The values listed will be crossed off the 10 by 10 grid of numbers. Something like 52 is crossed off since 4*13 = 52 is not square-free. The perfect square 4 is a factor.
Now let's look at the multiples of 9
9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108
Those values are crossed off the 10 by 10 grid of numbers for the same reasoning as the previous list.
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Keep this process going by crossing off multiples of perfect squares 16,25,36,49,64,81, and 100
This is what you should have leftover after those values are crossed off:
1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97
A number like 70 is square free because 70 = 2*5*7, none of those factors are a perfect square.
There are 61 square-free values in that list mentioned. This is out of 100 values total, so we get the fraction 61/100.
61/100 = 0.61 = 61%