Question 1119665
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<U>Answer</U>.  &nbsp;&nbsp;The smallest positive integer that has exactly &nbsp;6 &nbsp;divisors is &nbsp;&nbsp;12 = {{{2^2*3}}}.


<U>Solution</U>.


<pre>
1.  For integer number N = {{{p^alpha}}},  where p is a prime number and {{{alpha}}} is an integer exponent (index), 

    the number of divisors is {{{alpha + 1}}}.


    You can easily check it:  the divisors  are  1, p, {{{p^2}}}, . . . , {{{p^alpha}}}.



2.  For integer number  N = {{{p^alpha*q^beta*ellipsis*r^theta}}},  where p, q, . . . , r are prime divisors and {{{alpha}}}, {{{beta}}}, . . . , {{{theta}}} are integer exponents (indexes)  

    the number of divisors is  {{{(alpha+1)*(beta+1)*ellipsis*(theta+1)}}}.



3.  From these facts, you can easily obtain the answer.


    Notice that  6 = 2*3.


    It is easy to list those divisors:  1, 2, 4, 3, 6, 12.
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