Things that happen easily don't require a lot of effort. When your old car starts easily on a very cold morning, you'll breathe a sigh of relief.
While you can only climb a mountain with difficulty, you can probably climb the stairs in your house easily. You can also use this adverb to mean something closer to "quickly," as when you describe your puppy as easily distracted with treats or yourself as easily bored by long books. The Old French root of easily is aisie, "comfortable" or "at ease."
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