Cardinality set of multiples
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Given an arbitrarily large set of natural numbers greater than one,
S = {$p_0$, $p_1$, ... $p_n$}
product of S = $prod_{i=0}^n p_i$
define M as the set of all natural numbers that are multiples of any member of S, smaller or equal than the product and larger than zero.
Is there a formula for the cardinality of M given S?
E.g
S = {2,3}
M = {2,3,4,6}
cardinality of M = 4.
I can think of the formula for specific lengths but not of a general form.
multiplicative-function
asked Nov 22 at 9:26
JFugger_jr
93
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up vote
2
down vote
favorite
Given an arbitrarily large set of natural numbers greater than one,
S = {$p_0$, $p_1$, ... $p_n$}
product of S = $prod_{i=0}^n p_i$
define M as the set of all natural numbers that are multiples of any member of S, smaller or equal than the product and larger than zero.
Is there a formula for the cardinality of M given S?
E.g
S = {2,3}
M = {2,3,4,6}
cardinality of M = 4.
I can think of the formula for specific lengths but not of a general form.
multiplicative-function
asked Nov 22 at 9:26
JFugger_jr
93
New contributor
JFugger_jr is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Given an arbitrarily large set of natural numbers greater than one,
S = {$p_0$, $p_1$, ... $p_n$}
product of S = $prod_{i=0}^n p_i$
define M as the set of all natural numbers that are multiples of any member of S, smaller or equal than the product and larger than zero.
Is there a formula for the cardinality of M given S?
E.g
S = {2,3}
M = {2,3,4,6}
cardinality of M = 4.
I can think of the formula for specific lengths but not of a general form.
multiplicative-function
asked Nov 22 at 9:26
JFugger_jr
93
New contributor
JFugger_jr is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Given an arbitrarily large set of natural numbers greater than one,
S = {$p_0$, $p_1$, ... $p_n$}
product of S = $prod_{i=0}^n p_i$
define M as the set of all natural numbers that are multiples of any member of S, smaller or equal than the product and larger than zero.
Is there a formula for the cardinality of M given S?
E.g
S = {2,3}
M = {2,3,4,6}
cardinality of M = 4.
I can think of the formula for specific lengths but not of a general form.
multiplicative-function
multiplicative-function
asked Nov 22 at 9:26
JFugger_jr
93
New contributor
JFugger_jr is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Nov 22 at 9:26
JFugger_jr
93
New contributor
JFugger_jr is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Nov 22 at 9:26
JFugger_jr
93
New contributor
JFugger_jr is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Nov 22 at 9:26
JFugger_jr
93
asked Nov 22 at 9:26
JFugger_jr
93
93
New contributor
JFugger_jr is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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New contributor
JFugger_jr is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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1 Answer
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Let us consider all numbers from $1$ to $N=prod_k p_k$ identified with the elements of the ring $Bbb Z/Ncongprod_kBbb Z/p_k$ of integers modulo $N$. Then the elements divisible by one of the factors, including zero (i.e. $N$ modulo $N$), are the non-units. The number of the units is given by the Euler indicator
$$phi(N)=N;prod_kleft(1-frac 1{p_k}right) .$$
Now consider "the complement", to get the answer $N-phi(N)$. For example, for $N=6$ we get $6-phi(6)=6-2=4$.
answered Nov 22 at 10:03
dan_fulea
5,9751312
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Let us consider all numbers from $1$ to $N=prod_k p_k$ identified with the elements of the ring $Bbb Z/Ncongprod_kBbb Z/p_k$ of integers modulo $N$. Then the elements divisible by one of the factors, including zero (i.e. $N$ modulo $N$), are the non-units. The number of the units is given by the Euler indicator
$$phi(N)=N;prod_kleft(1-frac 1{p_k}right) .$$
Now consider "the complement", to get the answer $N-phi(N)$. For example, for $N=6$ we get $6-phi(6)=6-2=4$.
answered Nov 22 at 10:03
dan_fulea
5,9751312
add a comment |
up vote
1
down vote
accepted
Let us consider all numbers from $1$ to $N=prod_k p_k$ identified with the elements of the ring $Bbb Z/Ncongprod_kBbb Z/p_k$ of integers modulo $N$. Then the elements divisible by one of the factors, including zero (i.e. $N$ modulo $N$), are the non-units. The number of the units is given by the Euler indicator
$$phi(N)=N;prod_kleft(1-frac 1{p_k}right) .$$
Now consider "the complement", to get the answer $N-phi(N)$. For example, for $N=6$ we get $6-phi(6)=6-2=4$.
answered Nov 22 at 10:03
dan_fulea
5,9751312
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Let us consider all numbers from $1$ to $N=prod_k p_k$ identified with the elements of the ring $Bbb Z/Ncongprod_kBbb Z/p_k$ of integers modulo $N$. Then the elements divisible by one of the factors, including zero (i.e. $N$ modulo $N$), are the non-units. The number of the units is given by the Euler indicator
$$phi(N)=N;prod_kleft(1-frac 1{p_k}right) .$$
Now consider "the complement", to get the answer $N-phi(N)$. For example, for $N=6$ we get $6-phi(6)=6-2=4$.
answered Nov 22 at 10:03
dan_fulea
5,9751312
Let us consider all numbers from $1$ to $N=prod_k p_k$ identified with the elements of the ring $Bbb Z/Ncongprod_kBbb Z/p_k$ of integers modulo $N$. Then the elements divisible by one of the factors, including zero (i.e. $N$ modulo $N$), are the non-units. The number of the units is given by the Euler indicator
$$phi(N)=N;prod_kleft(1-frac 1{p_k}right) .$$
Now consider "the complement", to get the answer $N-phi(N)$. For example, for $N=6$ we get $6-phi(6)=6-2=4$.
answered Nov 22 at 10:03
dan_fulea
5,9751312
answered Nov 22 at 10:03
dan_fulea
5,9751312
answered Nov 22 at 10:03
dan_fulea
5,9751312
answered Nov 22 at 10:03
dan_fulea
5,9751312
5,9751312
add a comment |
add a comment |
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