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 $4$.
${a_1 , a_2 } = { (1,6) (2,3) (3,2) (6,1) }$
number-theory
edited Nov 18 at 6:52
Joey Kilpatrick
1,183422
asked Nov 18 at 6:36
Tanu kumar
11
add a comment |
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
edited Nov 18 at 6:52
Joey Kilpatrick
1,183422
asked Nov 18 at 6:36
Tanu kumar
11
add a comment |
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
edited Nov 18 at 6:52
Joey Kilpatrick
1,183422
asked Nov 18 at 6:36
Tanu kumar
11
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
edited Nov 18 at 6:52
Joey Kilpatrick
1,183422
asked Nov 18 at 6:36
Tanu kumar
11
edited Nov 18 at 6:52
Joey Kilpatrick
1,183422
asked Nov 18 at 6:36
Tanu kumar
11
edited Nov 18 at 6:52
Joey Kilpatrick
1,183422
edited Nov 18 at 6:52
Joey Kilpatrick
1,183422
edited Nov 18 at 6:52
Joey Kilpatrick
1,183422
1,183422
asked Nov 18 at 6:36
Tanu kumar
11
asked Nov 18 at 6:36
Tanu kumar
11
asked Nov 18 at 6:36
Tanu kumar
11
11
add a comment |
add a comment |
1 Answer
1
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votes
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1
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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.
answered Nov 18 at 7:43
Empy2
33.3k12261
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
answered Nov 18 at 7:43
Empy2
33.3k12261
add a comment |
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.
answered Nov 18 at 7:43
Empy2
33.3k12261
add a comment |
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.
answered Nov 18 at 7:43
Empy2
33.3k12261
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.
answered Nov 18 at 7:43
Empy2
33.3k12261
answered Nov 18 at 7:43
Empy2
33.3k12261
answered Nov 18 at 7:43
Empy2
33.3k12261
answered Nov 18 at 7:43
Empy2
33.3k12261
33.3k12261
add a comment |
add a comment |
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