• It probably is that simple, but there is a high likelihood of the crank being deformed. Being aluminium and softer than the steel axle it has been wobbling on it will have distorted the taper that mates them together, so even if you get it tight it might well come loose again.
    You will then need a new crank.
    A bike shop without the puller is odd. The puller you will need to remove it when it is tight is one of the commoner tools. I am wondering if something else is up with it.


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