Question 1118682
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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.<br>
But that method will be very slow and inefficient for solving the kind of problem you have here; there are much better ways.<br>
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.<br>
Example 1: 72 = (2^3)(3^2); the number of positive divisors is (3+1)(2+1) = 4*3 = 12<br>
Example 2: 160 = (2^5)(5^1); the number of positive divisors is (5+1)(1+1) = 6*2 = 12<br>
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.<br>
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).<br>
But in your problem, where the number of positive divisors is 7, there is only one way to get that: 7 = (6+1).<br>
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.<br>
The smallest number of that form is 2^6 = 64.  So 64 is the smallest positive integer with 7 positive integer divisors.