Midrasch zu Schemuel I 2:11

Why do we recite twenty-four blessings on fast-days? Because that number corresponds to the twenty-four supplications uttered by Solomon as he brought the ark, while fasting, into the innermost part of the Sanctuary (the Holy of Holies). Why do we say nine benedictions on Rosh Hashanah? Rava the son of Hanina stated: To correspond to the nine times that Hannah mentioned the Divine Name in her prayer, commencing with the words and Hannah prayed (I Sam. 2:11).