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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.

JFugger_jr
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1 Answers1

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Let us consider all numbers from $1$ to $N=\prod_k p_k$ identified with the elements of the ring $\Bbb Z/N\cong\prod_k\Bbb Z/p_k$ 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 $$\phi(N)=N\;\prod_k\left(1-\frac 1{p_k}\right)\ .$$ Now consider "the complement", to get the answer $N-\phi(N)$. For example, for $N=6$ we get $6-\phi(6)=6-2=4$.

dan_fulea
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