I am onion!

Cody has 4 types of onions.

\(\bullet\) The number of \(\color{Purple}{Purple}\) onions can be any non-negative integer.

\(\bullet\) The number of \(\color{Green}{Green}\) onions is a multiple of 2.

\(\bullet\) The number of \(\color{Red}{Red}\) onions is a multiple of 3.

\(\bullet\) The number of \(\color{Blue}{Blue}\) onions is a multiple of 5.

If Cody has 23 onions, how many different distributions of colors can there be?

Details and Assumptions: Cody can't have a negative number of onions, but he may have 0 onions of a certain type(s).

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