gagee ganeun gime teipeurang memojido jom sada jullae?

-는 김에 means something like while you’re at it, since you’re doing that anyway, or on the occasion of doing that.

So 가게에 가는 김에 is not just while going to the store in a neutral time sense. It specifically suggests:

• the listener is already going to the store anyway
• because they are doing that, the speaker wants to add one more small favor

That is why this pattern often feels like Since you’re going anyway, could you also…?

Before 김에, the verb usually appears in a modifying form.

Here:

• 가다 = to go
• 가는 = the adnominal/modifying form of 가다

So 가는 김에 is the natural form for while going / since you’re going.

A learner-friendly way to think about it is:

• 김에 needs a verb form that can attach to a following noun-like element
• 가는 is the standard form here for an action that is happening or planned as the relevant occasion

You may also see past forms like:

• 온 김에 = since you came / while you’re here

So the form changes depending on the time reference.