布 is a compound ideograph: a hand (originally written 父 in this position, the ancient form of the hand component) plus 巾 (a cloth / kerchief). The composite reads as "a hand spreading out cloth" — the act of unfurling fabric. From this concrete action grew three semantic branches: spreading out → cloth itself (the thing spread) → distribution / publication (spreading information). The 巾 (cloth) radical anchors a small but unified family of fabric-and-flag characters: 帽 (hat), 帯 (sash), 席 (mat / seat), 常 (constant — originally a long banner).

Korean reading "po." 麻布 (mapo, hemp cloth — also a Seoul district name 마포구), 分布 (bunpo, distribution — used in geography, statistics, and biology), 公布 (gongpo, "public-spreading" = official promulgation, as of laws), 宣布 (seonpo, declaration / proclamation), 布陣 (pojin, military deployment / lineup — sports vocabulary too), 瀑布 (pokpo, "explosive cloth" = waterfall — note the spreading/falling fabric metaphor for cascading water).

Mandarin bù, 4th tone. 布 (bù, cloth / to spread / to declare), 布料 (bùliào, fabric — used in textile shops), 分布 (fēnbù, distribution), 宣布 (xuānbù, to announce), 瀑布 (pùbù, waterfall — same imagery as Korean), 公布 (gōngbù, to announce publicly). 布 also functions as a verb actively in modern Chinese: 布置 (bùzhì, to arrange / set up — what teachers do with classroom assignments), 摆布 (bǎibù, to arrange / manipulate).

Japanese on-reading フ (fu) — 分布 (bunpu, distribution), 宣布 (senpu, declaration), 公布 (kōfu, official promulgation). Kun-reading ぬの (nuno) — 布 (nuno, cloth / fabric — everyday word), 布巾 (fukin, dish towel). Most beloved Japanese compound: 布団 (futon, the famous quilted sleeping mattress — written with 布 + 団 "round"; the futon defines Japanese sleeping arrangements, a thin folded mat laid directly on tatami floors and stored away during the day). The English "futon" entered the language directly from Japanese 布団.

Memory aid: a hand spreading cloth (巾) — from the action of unfurling fabric came cloth itself, distribution, and announcement.