Asking questions, we are told, is one way that Korah challenged Moses. He asked: does a house full of sacred books require amezuzah?  It does. To the world, a mezuzahannounces that herein lies a Jewish home and this display of Jewish pride is important. The Temple, we are taught from a verse, did not have a mezuzah and this is why many don't affix a mezuzah to Synagogues. One may also suggest that Synagogues automatically proclaim to the world that herein lies a Jewish place.