Question 1129602
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<U>Solution 1</U>


<pre>
    The product of all positive divisors of a positive integer N is  6*16*36*96 = 331776.    (given)


    The product of all the positive divisors of the number 24 is  2*3*4*6*8*12*24 = 331776.


    <U>ANSWER</U>.  The number under the question is 24.
</pre>

Solved.


It is "experimental", or "empiric" prove.


The solution can be made with logical reasoning, too.



<U>Solution 2</U>


<pre>
    Notice that  6*16*36*96 = {{{2^12*3^4}}}.      (*)


    So, it is prime decomposition into the product of prime numbers 2 and 3 with their corresponding degrees.


    It implies that the "base" number  N  is the product of prime numbers 2 and 3 in some degrees :  N = {{{2^a*3^b}}}.     (**)


    From the other side,  if R is some positive integer divisor of N, then  S= N/R  is its other positive integer divisor - 
    - therefore, the product of all divisors of the integer N is N in some degree - namely in the degree, equal to half
    of the number of all its divisors.


    Thus the number (*)  is some degree of N, and looking into formula (**), you may hypothesize than the degree b is 4.


    Then {{{2^12}}} must be the 4-th degree, and then it  <U>necessary</U>  is {{{(2^3)^4}}}, so the number N is necessary  N = {{{2^3*3}}} = 8*3 = 24,

    exactly as we found it in the first,  "experimental", or "empiric" proof.



    Notice that  {{{2^12*3^4}}} = {{{24^4}}}  and that the number 24 has 8 divisors  1, 2, 3, 4, 6, 8, 12 and 24.
</pre>

Solved twice and explained.