der Schlüssel
key
A metal tool used to open or close a lock.
Ich glaube, ich habe meinen Schlüssel in der Wohnung vergessen.
I think I forgot my key in the apartment.

key / wrench
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key
A metal tool used to open or close a lock.
Ich glaube, ich habe meinen Schlüssel in der Wohnung vergessen.
I think I forgot my key in the apartment.

'Der Schlüssel' (the key) is closely related to the verb 'schließen' (to close/lock) and the noun 'das Schloss' (the lock). With a key, you can close/lock or open a lock.
Ich kann die Tür nicht schließen, das Schloss ist kaputt.
I can't close the door; the lock is broken.
In the figurative meaning 'Lösung' (solution), 'Schlüssel' (key) is almost always used with the preposition 'zu' and the dative. It is the key to something.
Geduld ist der Schlüssel zum Erfolg.
Patience is the key to success.
For a door key or car key, compound words are often used that describe the purpose more precisely, e.g. 'Haustürschlüssel' (front door key), 'Wohnungsschlüssel' (apartment key), or 'Autoschlüssel' (car key).
Hast du den Ersatzschlüssel für das Auto gesehen?
Have you seen the spare key for the car?
The meaning 'Werkzeug' (tool) can be recognized from the context of crafts, repairs, or technology. Often the size is added as a number, e.g. 'ein 10er-Schlüssel' (a 10-millimeter wrench).
Für diese Schraube brauchst du einen anderen Schlüssel.
You need a different wrench for this screw.
In the meaning 'Verteilungsformel' (allocation formula), the phrasing 'nach einem Schlüssel' (according to a formula/key) or 'gemäß einem Schlüssel' (in accordance with a formula/key) is often used. This is typical in administrative or financial contexts.
Die Plätze werden nach einem komplizierten Schlüssel vergeben.
The places are allocated according to a complicated formula.
The meaning 'Code' (code) is common in the digital field. Words like 'digital' (digital), 'privat' (private), 'öffentlich' (public), 'Datei' (file), or 'verschlüsseln' (to encrypt) show that this kind of key is meant.
Sie müssen den privaten Schlüssel sicher aufbewahren.
You must keep the private key safe.
In a musical context, you can recognize this meaning from words like 'Noten' (notes), 'Notensystem' (staff), 'Violine' (violin), or 'Bass' (bass). The most common are the 'Violinschlüssel' (treble clef) and the 'Bassschlüssel' (bass clef).
Hohe Instrumente wie die Flöte werden im Violinschlüssel notiert.
High instruments such as the flute are notated in the treble clef.
solution
'die Lösung' and 'der Schlüssel' can both describe how you understand or solve a problem. 'die Lösung' is directly the answer or the way out of a problem. 'der Schlüssel' is often used figuratively and refers to the important point that leads to the solution. You can say: the key helps you find the solution. The words can be confused because both appear with tasks, riddles, or problems. If you mean the result, 'die Lösung' fits better. If you mean the decisive access point or clue, 'der Schlüssel' sounds more natural.

Geduld ist bei dieser Arbeit der Schlüssel zum Erfolg.
Patience is the key to success in this work.
Für die Aufgabe findet Maria nach langem Rechnen endlich die Lösung.
After a lot of calculating, Maria finally finds the solution to the task.