Find the smallest value of positive integer !!












1












$begingroup$


Find the smallest positive integer with exactly 30 positive factor



First, I use function $tau$ to find the exponential that gives $2×3×5$ and I want to find the smallest value.
How can I find it use inequality to help



And how to find the value of $sqrt{8×13×15×17+49}$? I get $x=8$ and doing perfect square ... but stuck who can help me, please






arithmetic integers







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$endgroup$








  • 1




    $begingroup$
    Your question is a bit unclear. Can you give some examples of the smallest value of a positive number with 2,3,4 positive factors? What do you mean with positive factors?
    $endgroup$
    – kvantour
    Jan 31 at 13:11






  • 1




    $begingroup$
    What have you tried? Given the primne factorization $n=prod p_i^{a_i}$ , do you know how to compute the number of factors of $n$?
    $endgroup$
    – lulu
    Jan 31 at 13:11










  • $begingroup$
    @kvantour To me these feel like two separate questions that should have been asked separately, in my opinion.
    $endgroup$
    – Robert Soupe
    Jan 31 at 18:39


















1












$begingroup$


Find the smallest positive integer with exactly 30 positive factor



First, I use function $tau$ to find the exponential that gives $2×3×5$ and I want to find the smallest value.
How can I find it use inequality to help



And how to find the value of $sqrt{8×13×15×17+49}$? I get $x=8$ and doing perfect square ... but stuck who can help me, please






arithmetic integers







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Your question is a bit unclear. Can you give some examples of the smallest value of a positive number with 2,3,4 positive factors? What do you mean with positive factors?
    $endgroup$
    – kvantour
    Jan 31 at 13:11






  • 1




    $begingroup$
    What have you tried? Given the primne factorization $n=prod p_i^{a_i}$ , do you know how to compute the number of factors of $n$?
    $endgroup$
    – lulu
    Jan 31 at 13:11










  • $begingroup$
    @kvantour To me these feel like two separate questions that should have been asked separately, in my opinion.
    $endgroup$
    – Robert Soupe
    Jan 31 at 18:39
















1












1








1





$begingroup$


Find the smallest positive integer with exactly 30 positive factor



First, I use function $tau$ to find the exponential that gives $2×3×5$ and I want to find the smallest value.
How can I find it use inequality to help



And how to find the value of $sqrt{8×13×15×17+49}$? I get $x=8$ and doing perfect square ... but stuck who can help me, please






arithmetic integers







share|cite|improve this question











$endgroup$




Find the smallest positive integer with exactly 30 positive factor



First, I use function $tau$ to find the exponential that gives $2×3×5$ and I want to find the smallest value.
How can I find it use inequality to help



And how to find the value of $sqrt{8×13×15×17+49}$? I get $x=8$ and doing perfect square ... but stuck who can help me, please






arithmetic integers




arithmetic integers






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 31 at 15:53









Larry

2,53031131




2,53031131










asked Jan 31 at 13:07









HeartHeart

30519




30519








  • 1




    $begingroup$
    Your question is a bit unclear. Can you give some examples of the smallest value of a positive number with 2,3,4 positive factors? What do you mean with positive factors?
    $endgroup$
    – kvantour
    Jan 31 at 13:11






  • 1




    $begingroup$
    What have you tried? Given the primne factorization $n=prod p_i^{a_i}$ , do you know how to compute the number of factors of $n$?
    $endgroup$
    – lulu
    Jan 31 at 13:11










  • $begingroup$
    @kvantour To me these feel like two separate questions that should have been asked separately, in my opinion.
    $endgroup$
    – Robert Soupe
    Jan 31 at 18:39
















  • 1




    $begingroup$
    Your question is a bit unclear. Can you give some examples of the smallest value of a positive number with 2,3,4 positive factors? What do you mean with positive factors?
    $endgroup$
    – kvantour
    Jan 31 at 13:11






  • 1




    $begingroup$
    What have you tried? Given the primne factorization $n=prod p_i^{a_i}$ , do you know how to compute the number of factors of $n$?
    $endgroup$
    – lulu
    Jan 31 at 13:11










  • $begingroup$
    @kvantour To me these feel like two separate questions that should have been asked separately, in my opinion.
    $endgroup$
    – Robert Soupe
    Jan 31 at 18:39










1




1




$begingroup$
Your question is a bit unclear. Can you give some examples of the smallest value of a positive number with 2,3,4 positive factors? What do you mean with positive factors?
$endgroup$
– kvantour
Jan 31 at 13:11




$begingroup$
Your question is a bit unclear. Can you give some examples of the smallest value of a positive number with 2,3,4 positive factors? What do you mean with positive factors?
$endgroup$
– kvantour
Jan 31 at 13:11




1




1




$begingroup$
What have you tried? Given the primne factorization $n=prod p_i^{a_i}$ , do you know how to compute the number of factors of $n$?
$endgroup$
– lulu
Jan 31 at 13:11




$begingroup$
What have you tried? Given the primne factorization $n=prod p_i^{a_i}$ , do you know how to compute the number of factors of $n$?
$endgroup$
– lulu
Jan 31 at 13:11












$begingroup$
@kvantour To me these feel like two separate questions that should have been asked separately, in my opinion.
$endgroup$
– Robert Soupe
Jan 31 at 18:39






$begingroup$
@kvantour To me these feel like two separate questions that should have been asked separately, in my opinion.
$endgroup$
– Robert Soupe
Jan 31 at 18:39












2 Answers
2






active

oldest

votes


















1












$begingroup$

The number of factors of a natural is given by the product of the multiplicities of its prime factors plus one, $(m_2+1)(m_3+1)(m_5+1)cdots$.



To obtain $30=2cdot3cdot5$ factors, you need multiplicities $1,2$ and $4$, which you will assign to the smallest possible primes, by decreasing order



$$5^13^22^4=720.$$






share|cite|improve this answer









$endgroup$





















    0












    $begingroup$

    What do you mean by "exactly 5 positive factors"? Are these distinct or can they be the same? If the same then the smallest such integer is $2^5= 32$. It they must all be different then it is 2(3)(4)(5)(6)= 720.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      The question about distinctness doesn't make sense. $1$ has thirty factors $1$, and many more.
      $endgroup$
      – Yves Daoust
      Jan 31 at 13:22










    • $begingroup$
      Because of that "1" is not usually considered a factor.
      $endgroup$
      – user247327
      Jan 31 at 13:33










    • $begingroup$
      Don't you understand ? $2$ has thirty factors $2$, and many more.
      $endgroup$
      – Yves Daoust
      Jan 31 at 15:34












    Your Answer





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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    1












    $begingroup$

    The number of factors of a natural is given by the product of the multiplicities of its prime factors plus one, $(m_2+1)(m_3+1)(m_5+1)cdots$.



    To obtain $30=2cdot3cdot5$ factors, you need multiplicities $1,2$ and $4$, which you will assign to the smallest possible primes, by decreasing order



    $$5^13^22^4=720.$$






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      The number of factors of a natural is given by the product of the multiplicities of its prime factors plus one, $(m_2+1)(m_3+1)(m_5+1)cdots$.



      To obtain $30=2cdot3cdot5$ factors, you need multiplicities $1,2$ and $4$, which you will assign to the smallest possible primes, by decreasing order



      $$5^13^22^4=720.$$






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        The number of factors of a natural is given by the product of the multiplicities of its prime factors plus one, $(m_2+1)(m_3+1)(m_5+1)cdots$.



        To obtain $30=2cdot3cdot5$ factors, you need multiplicities $1,2$ and $4$, which you will assign to the smallest possible primes, by decreasing order



        $$5^13^22^4=720.$$






        share|cite|improve this answer









        $endgroup$



        The number of factors of a natural is given by the product of the multiplicities of its prime factors plus one, $(m_2+1)(m_3+1)(m_5+1)cdots$.



        To obtain $30=2cdot3cdot5$ factors, you need multiplicities $1,2$ and $4$, which you will assign to the smallest possible primes, by decreasing order



        $$5^13^22^4=720.$$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 31 at 13:28









        Yves DaoustYves Daoust

        132k676230




        132k676230























            0












            $begingroup$

            What do you mean by "exactly 5 positive factors"? Are these distinct or can they be the same? If the same then the smallest such integer is $2^5= 32$. It they must all be different then it is 2(3)(4)(5)(6)= 720.






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              The question about distinctness doesn't make sense. $1$ has thirty factors $1$, and many more.
              $endgroup$
              – Yves Daoust
              Jan 31 at 13:22










            • $begingroup$
              Because of that "1" is not usually considered a factor.
              $endgroup$
              – user247327
              Jan 31 at 13:33










            • $begingroup$
              Don't you understand ? $2$ has thirty factors $2$, and many more.
              $endgroup$
              – Yves Daoust
              Jan 31 at 15:34
















            0












            $begingroup$

            What do you mean by "exactly 5 positive factors"? Are these distinct or can they be the same? If the same then the smallest such integer is $2^5= 32$. It they must all be different then it is 2(3)(4)(5)(6)= 720.






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              The question about distinctness doesn't make sense. $1$ has thirty factors $1$, and many more.
              $endgroup$
              – Yves Daoust
              Jan 31 at 13:22










            • $begingroup$
              Because of that "1" is not usually considered a factor.
              $endgroup$
              – user247327
              Jan 31 at 13:33










            • $begingroup$
              Don't you understand ? $2$ has thirty factors $2$, and many more.
              $endgroup$
              – Yves Daoust
              Jan 31 at 15:34














            0












            0








            0





            $begingroup$

            What do you mean by "exactly 5 positive factors"? Are these distinct or can they be the same? If the same then the smallest such integer is $2^5= 32$. It they must all be different then it is 2(3)(4)(5)(6)= 720.






            share|cite|improve this answer









            $endgroup$



            What do you mean by "exactly 5 positive factors"? Are these distinct or can they be the same? If the same then the smallest such integer is $2^5= 32$. It they must all be different then it is 2(3)(4)(5)(6)= 720.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Jan 31 at 13:14









            user247327user247327

            11.5k1516




            11.5k1516












            • $begingroup$
              The question about distinctness doesn't make sense. $1$ has thirty factors $1$, and many more.
              $endgroup$
              – Yves Daoust
              Jan 31 at 13:22










            • $begingroup$
              Because of that "1" is not usually considered a factor.
              $endgroup$
              – user247327
              Jan 31 at 13:33










            • $begingroup$
              Don't you understand ? $2$ has thirty factors $2$, and many more.
              $endgroup$
              – Yves Daoust
              Jan 31 at 15:34


















            • $begingroup$
              The question about distinctness doesn't make sense. $1$ has thirty factors $1$, and many more.
              $endgroup$
              – Yves Daoust
              Jan 31 at 13:22










            • $begingroup$
              Because of that "1" is not usually considered a factor.
              $endgroup$
              – user247327
              Jan 31 at 13:33










            • $begingroup$
              Don't you understand ? $2$ has thirty factors $2$, and many more.
              $endgroup$
              – Yves Daoust
              Jan 31 at 15:34
















            $begingroup$
            The question about distinctness doesn't make sense. $1$ has thirty factors $1$, and many more.
            $endgroup$
            – Yves Daoust
            Jan 31 at 13:22




            $begingroup$
            The question about distinctness doesn't make sense. $1$ has thirty factors $1$, and many more.
            $endgroup$
            – Yves Daoust
            Jan 31 at 13:22












            $begingroup$
            Because of that "1" is not usually considered a factor.
            $endgroup$
            – user247327
            Jan 31 at 13:33




            $begingroup$
            Because of that "1" is not usually considered a factor.
            $endgroup$
            – user247327
            Jan 31 at 13:33












            $begingroup$
            Don't you understand ? $2$ has thirty factors $2$, and many more.
            $endgroup$
            – Yves Daoust
            Jan 31 at 15:34




            $begingroup$
            Don't you understand ? $2$ has thirty factors $2$, and many more.
            $endgroup$
            – Yves Daoust
            Jan 31 at 15:34


















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