Jesus' quotation of Septuagint - Greek or Aramaic?

Having read this and this , I am convinced now that Jesus spoke Aramaic. However, I also know that many times He was quoting Septuagint, which is a translation of the Old Testament into Koine Greek. I am a little bit confused now. Does it mean that when He was talking to His disciples or to the Jews and was quoting Septuagint, He was switching from Aramaic to Greek? If yes, then it sounds a bit contradictory to me. Why would He need to switch to Greek while He was talking to the Hebrew people, not to the Greeks, about their own Holy Scripture (Jews' aversion toward the Greek is quite evident from the book of Acts)? If no, then how can we say that He was quoting Septuagint (as I know Septuagint was written only in Greek)?