Q. After the 40 days of fasting, Jesus is tempted. In His response to Satan, why does Jesus quote from Deuteronomy and not another book and why are all three responses from just the one? Is there more here that I’m not getting?
This is an excellent and very perceptive question. When Jesus was baptized, a voice from heaven said, “This is my Son, whom I love; with him I am well pleased.” Most interpreters take this to be an announcement that Jesus was the Messiah, and they understand this to be the moment when what had been a growing realization crystallized for Jesus that he was the Messiah. He immediately went alone into the wilderness to understand what the implications of this were.
In the temptations, the devil was basically saying to Jesus, “So, I hear you’re the Messiah. That’s great. Have you thought about what kind of Messiah you’re going to be?” (“If you are the Son of God …”) The temptations were to see his primary role as that of meeting the physical needs of people; to do dazzling daredevil feats that would win admiration and an audience; or to try to achieve his purposes by obtaining political and military power. Interestingly, later on Jesus actually did feed people miraculously, and on many occasions he was delivered spectacularly from dangers, although he definitively rejected pursuing political and military power.
But on this occasion, it would have been wrong to do any of those things as primary to his Messianic vocation, particularly at the suggestion of the devil that this was the kind of Messiah he should be. And just as interestingly, Jesus rejects all of these temptations on the basis that they would involve doing something that would be wrong for anyone to do—seeing life as consisting primarily of meeting physical needs and desires; putting God to the test; worshiping anyone but God. In each case, Jesus cites scriptures to show that this would be wrong.
It makes sense to me that all of these scriptures would come from the Torah or law of Moses, because that is where the normative principles for godly conduct are stated directly in the Hebrew Bible. Indeed, while Jesus spoke a few times of “Moses and the prophets,” and once of “Moses and the prophets and the Psalms” (“Psalms” likely meaning the final section of the Hebrew Bible, the “Writings”), in general he spoke and taught about “the law” or “what Moses wrote” or “the law of Moses.” Principles for godly conduct can be inferred from the narratives, songs, etc. in other parts of the Bible, but they are laid out directly in the law of Moses.
That said, is there a reason why Jesus would have quoted all of these principles specifically from the book of Deuteronomy, rather than from some other book of the Torah? We could say that it was simply a coincidence that they were all found there. But perhaps, as you say, there is something more going on here.
Deuteronomy is a single long discourse by Moses. In the gospel of Matthew, the temptations are followed by the Sermon on the Mount, a single long discourse by Jesus, in which he explains the deepest meanings and applications of the law. In Luke, the equivalent Sermon on the Plain comes not long after the temptations. So perhaps we are to understand what Jesus does in those discourses as an echo of Deuteronomy.
Matthew in particular portrays Jesus as a “new Moses” in many ways in his gospel, that is, as someone who will be a teacher and giver of a law that brings freedom. We may actually see Jesus entering into his vocation as this “new Moses” in the temptations themselves, as he articulates the meaning and application of “what Moses wrote” for the situations that the devil is describing. This would be a delightful irony. In the process of trying to get Jesus to be the wrong kind of Messiah, the devil provides the occasion for Jesus to step into his vocation as the right kind of Messiah. As that happens, the farewell speech of the first Moses provides the inaugural content for the new Moses.
Good Question 
Thank you. How interesting!
Andy