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I know いくつ means 'how old', but what does いくつか mean in the following sentence?

緑茶、紅茶、ウーロン茶などいくつかの種類に分けられる。

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いくつ in general means "how many"; this can also be applied to years to ask "how old" ("how many years") as you've pointed out.

か can be add to counter question words to mean "several ~".

  • ラーメン屋に学生が何人かいます。 → There are several (people) students in the ramen restaurant.
  • 先週のお祭りに何個かのたこ焼きを食った。 → I ate several takoyaki (balls) at the festival last week.
  • 緑茶、紅茶、ウーロン茶などいくつかの種類に分けられる。 → Tea is divided into several kinds, such as green tea, black tea, and oolong tea.
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いくつ can also be used as an interrogative for indefinite number, that is, how many. To use indefinite expressions in non-question, か is combined with it.

  • いくつか - some number
    • いくつかの種類 - some number of kinds
  • いつか - at some point (Of time)
    • いつか彼には知らせるつもりだった。- I was going to tell him at some point.
  • なぜか - for some reason
    • 彼女はなぜかこの頃私を避けてる。 - She is avoiding me for some reason.
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I know いくつ means how old

Without context, 「いくつ」 means "how many" far more often than it means "how old".

「いくつか」 means "some", "a few", "several", etc.

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