Sherlock Holmes NEVER said "Elementary, my dear Watson."
In Doyle's own writings, then, the bit of dialog that comes closest to matching the famous Holmesian signature phrase is this exchange between Holmes andDr. Watson in the short story "The Crooked Man":
"I see that you are professionally rather busy just now," said he, glancing very keenly across at me.
"Yes, I've had a busy day," I answered. "It may seem very foolish in your eyes," I added, "but really I don't know how you deduced it."
Holmes chuckled to himself.
"I have the advantage of knowing your habits, my dear Watson," said he. "When your round is a short one you walk, and when it is a long one you use a hansom. As I perceive that your boots, although used, are by no means dirty, I cannot doubt that you are at present busy enough to justify the hansom."
"Excellent!" I cried.
"Elementary," said he.