Question 1118682: What is the smallest positive that has 7 positive divisors?
Found 2 solutions by addingup, greenestamps:
Answer by addingup(3677)   (Show Source):
You can put this solution on YOUR website! Make a table with three columns and plug in numbers until you get to a number with 7 divisors, like this:
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Number - - List of divisors - - Number of divisors per number
------ . . ---------------- . . -------------------------
1 . . . . . . . 1 . . . . . . . . . . .1
2 . . . . . . . 2 . . . . . . . . . . . 1, 2
3 . . . . . . . 2 . . . . . . . . . . . 1, 3
4 . . . . . . . 3 . . . . . . . . . . . 1, 2, 4
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This is a wonderful exercise for you in learning the divisors and number of divisors for each number. Finding the LCD is one example of the practical application of this exercise.
Answer by greenestamps(13198)  (Show Source):
You can put this solution on YOUR website!
Looking at all the numbers starting with 1 and making a list of the divisors of each of them, as the other tutor suggests, can be a useful exercise in discovering patterns.
But that method will be very slow and inefficient for solving the kind of problem you have here; there are much better ways.
To find the number of divisors of an integer, the process is as follows:
(1) write the number in its prime factorization
(2) add 1 to each of the exponents in the prime factorization and multiply the resulting numbers; that is the number of positive divisors of the integer.
Example 1: 72 = (2^3)(3^2); the number of positive divisors is (3+1)(2+1) = 4*3 = 12
Example 2: 160 = (2^5)(5^1); the number of positive divisors is (5+1)(1+1) = 6*2 = 12
In your problem, you are given the number of positive divisors and you are asked to find the smallest positive integer with that number of divisors.
In the two examples above, both numbers had 12 positive divisors. That number was obtained differently for the two examples; in one example it was 12 = (3+1)(2+1); in the other it was 12 = (5+1)(1+1).
But in your problem, where the number of positive divisors is 7, there is only one way to get that: 7 = (6+1).
That means the only positive integers with 7 positive integer divisors are those in which one prime factor is to the 6th power, and there are no other prime factors -- i.e., numbers of the form p^6, where p is a prime number.
The smallest number of that form is 2^6 = 64. So 64 is the smallest positive integer with 7 positive integer divisors.
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