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You can put this solution on YOUR website!
\n" ); document.write( "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.
\n" ); document.write( "But that method will be very slow and inefficient for solving the kind of problem you have here; there are much better ways.
\n" ); document.write( "To find the number of divisors of an integer, the process is as follows:
\n" ); document.write( "(1) write the number in its prime factorization
\n" ); document.write( "(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.
\n" ); document.write( "Example 1: 72 = (2^3)(3^2); the number of positive divisors is (3+1)(2+1) = 4*3 = 12
\n" ); document.write( "Example 2: 160 = (2^5)(5^1); the number of positive divisors is (5+1)(1+1) = 6*2 = 12
\n" ); document.write( "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.
\n" ); document.write( "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).
\n" ); document.write( "But in your problem, where the number of positive divisors is 7, there is only one way to get that: 7 = (6+1).
\n" ); document.write( "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.
\n" ); document.write( "The smallest number of that form is 2^6 = 64. So 64 is the smallest positive integer with 7 positive integer divisors. \n" ); document.write( "