Etymology
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quire (n.1)

c. 1200, quaier, "a short book;" mid-15c., "a set of four folded pages for a book; pamphlet consisting of a single quire," original senses now obsolete, from Anglo-French quier, Old French quaier, caier "sheet of paper folded in four" (Modern French cahier), from Medieval Latin quaternum, "set of four sheets of parchment or paper," from Vulgar Latin *quaternus, from Latin quaterni "four each," from quater "four times" (from PIE root *kwetwer- "four").

Meaning "standard unit for selling paper" (lately typically 24 or 25 sheets, the twentieth part of a ream) is recorded from late 14c. In quires (mid-15c.) means "unbound."

quire (n.2)

an early form and later variant spelling of choir (q.v.), Middle English, from Old French quer, queor, variants of cuer, and compare Medieval Latin quorus, variant of chorus.

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Definitions of quire

quire (n.)
a quantity of paper; 24 or 25 sheets;
From wordnet.princeton.edu