How many sequences a of length $N$ consisting of positive integers satisfy $a_1 times a_2 times … times a_n...











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You are given positive integers $N$ and $M$ .



How many sequences a of length N consisting of positive integers satisfy $a_1 times a_2 times ... times a_n = M$ ?



Here , two sequences $a'$ and $a''$ are considered different when there exists some $i$ such that $ai' != ai''$ .



For example $N = 2$ and $M = 6$, the answer is $4$.



${a_1 , a_2 } = { (1,6) (2,3) (3,2) (6,1) }$






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    You are given positive integers $N$ and $M$ .



    How many sequences a of length N consisting of positive integers satisfy $a_1 times a_2 times ... times a_n = M$ ?



    Here , two sequences $a'$ and $a''$ are considered different when there exists some $i$ such that $ai' != ai''$ .



    For example $N = 2$ and $M = 6$, the answer is $4$.



    ${a_1 , a_2 } = { (1,6) (2,3) (3,2) (6,1) }$






    number-theory







    share|cite|improve this question


























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      You are given positive integers $N$ and $M$ .



      How many sequences a of length N consisting of positive integers satisfy $a_1 times a_2 times ... times a_n = M$ ?



      Here , two sequences $a'$ and $a''$ are considered different when there exists some $i$ such that $ai' != ai''$ .



      For example $N = 2$ and $M = 6$, the answer is $4$.



      ${a_1 , a_2 } = { (1,6) (2,3) (3,2) (6,1) }$






      number-theory







      share|cite|improve this question















      You are given positive integers $N$ and $M$ .



      How many sequences a of length N consisting of positive integers satisfy $a_1 times a_2 times ... times a_n = M$ ?



      Here , two sequences $a'$ and $a''$ are considered different when there exists some $i$ such that $ai' != ai''$ .



      For example $N = 2$ and $M = 6$, the answer is $4$.



      ${a_1 , a_2 } = { (1,6) (2,3) (3,2) (6,1) }$






      number-theory




      number-theory






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      share|cite|improve this question













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      edited Nov 18 at 6:52









      Joey Kilpatrick

      1,163121




      1,163121










      asked Nov 18 at 6:36









      Tanu kumar

      11




      11






















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          The different prime factors are independent.

          In your example, the 2 could appearas $(1,2)$ or $(2,1)$, and the 3 could be $(1,3)$ or $(3,1)$. Multiply them together, for example $(1,2)times(1,3)=(1times1,2times3)=(1,6)$.

          Now how many ways can $p^k$ be placed in $N$ spots? This is a good use of stars and bars.






          share|cite|improve this answer





















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            The different prime factors are independent.

            In your example, the 2 could appearas $(1,2)$ or $(2,1)$, and the 3 could be $(1,3)$ or $(3,1)$. Multiply them together, for example $(1,2)times(1,3)=(1times1,2times3)=(1,6)$.

            Now how many ways can $p^k$ be placed in $N$ spots? This is a good use of stars and bars.






            share|cite|improve this answer

























              up vote
              1
              down vote













              The different prime factors are independent.

              In your example, the 2 could appearas $(1,2)$ or $(2,1)$, and the 3 could be $(1,3)$ or $(3,1)$. Multiply them together, for example $(1,2)times(1,3)=(1times1,2times3)=(1,6)$.

              Now how many ways can $p^k$ be placed in $N$ spots? This is a good use of stars and bars.






              share|cite|improve this answer























                up vote
                1
                down vote










                up vote
                1
                down vote









                The different prime factors are independent.

                In your example, the 2 could appearas $(1,2)$ or $(2,1)$, and the 3 could be $(1,3)$ or $(3,1)$. Multiply them together, for example $(1,2)times(1,3)=(1times1,2times3)=(1,6)$.

                Now how many ways can $p^k$ be placed in $N$ spots? This is a good use of stars and bars.






                share|cite|improve this answer












                The different prime factors are independent.

                In your example, the 2 could appearas $(1,2)$ or $(2,1)$, and the 3 could be $(1,3)$ or $(3,1)$. Multiply them together, for example $(1,2)times(1,3)=(1times1,2times3)=(1,6)$.

                Now how many ways can $p^k$ be placed in $N$ spots? This is a good use of stars and bars.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Nov 18 at 7:43









                Empy2

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