רמך Royal mares
Every verse in which this word stands — 1 place, in canonical order, first appearance to last. These are the places the lexicon carries under this word's own entry, counting every form the text writes it in, which is the number on the list.
Once, in Esther 8:10 — the couriers ride 'sons of the ramachim.' Traditionally brood mares of the royal stud; the identification is discussed.
וַיִּכְתֹּב בְּשֵׁם הַמֶּלֶךְ אֲחַשְׁוֵרֹשׁ וַיַּחְתֹּם בְּטַבַּעַת הַמֶּלֶךְ וַיִּשְׁלַח סְפָרִים בְּיַד הָרָצִים בַּסּוּסִים רֹכְבֵי הָרֶכֶשׁ הָֽאֲחַשְׁתְּרָנִים בְּנֵי הָֽרַמָּכִֽים
And they wrote in the name of king Ahasuerus, and sealed it with the king’s ring, and sent letters by posts on horseback, riding on swift steeds that were used in the king’s service, bred of the stud;
Quick answers
What is the Hebrew word for royal mares?
Royal mares in Hebrew is רמך, with a gematria of 260 in the standard method.
How many times is the royal mares mentioned in the Tanakh?
The word stands 1 time in the Tanakh, first at Esther 8:10.
A spelling is not a word. The corpus never writes רמך bare — every one of its places carries a particle or a suffix on the front or the back — so the spelling above is the word's own headword, and the verses listed here are the only door to it.