ccassel wrote:Thanks.
How would you answer the questions if the obvious plug-in was not available?
To determine the number of positive factors of an integer:
1) Prime-factorize the integer
2) Add 1 to each exponent
3) Multiply
For example:
72 = 2³ * 3².
Adding 1 to each exponent and multiplying, we get (3+1)*(2+1) = 12 factors.
Here's the reasoning. To determine how many factors can be created from 72 = 2³ * 3², we need to determine the number of choices we have of each prime factor:
For 2, we can use 2�, 2¹, 2², or 2³, giving us 4 choices.
For 3, we can use 3�, 3¹, or 3², giving us 3 choices.
Multiplying, we get 4*3 = 12 possible factors.
Thus, to count the total number of divisors of n, we would prime-factorize n, add 1 to each exponent, and multiply.
Thus, if n has exactly 3 divisors, we know that n = (prime number)². (Because if we add 1 to the exponent of 2, we'll get 2+1 = 3 factors.)
Thus, n² = (prime number²)² = (prime number)�.
Adding 1 to the exponent, we know that n² must have 4+1 = 5 factors.
I think that plugging in -- as I did in my original post -- is much easier.
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