Origin Story
Amu-ryeom is a shortened form of amu-ryeomyeon. Amu-ryeomyeon means "however that may be," and unpacked further, it carries the conviction that "no matter what other cases you imagine, in the end it is naturally so." The expression was used to show deep agreement with what someone said, or when a fact was so obvious it hardly needed stating. It was often used as a rhetorical question — "Amu-ryeomyeon, wouldn't it be so?" — before gradually hardening into the short interjection amu-ryeom. Into this single word are folded the warm agreement and certainty of "I think so too" and "Of course."
A word built on a similar structure is amu-raedo. A contraction of amu-ri haedo ("no matter how much you try"), it means "however you think about it, in the end..." Both spring from amu-ri and branched off into siblings with different shades of meaning.
Meaning Evolution
How It Is Used
Of course ("amu-ryeom"), you're right.
Absolutely, no doubt about it. Of course I'll help.
Ah well, what does it matter ("amu-ryeom eotteora") — it's already water under the bridge.
Related Words
Memory Hook
Remember "amu-ryeomyeon (however that may be) → amu-ryeom": whatever you think, it's obvious — that's amu-ryeom.
"In what we take for granted lies the deepest trust of all."