Do 2 Kings 15:30 and 2 Kings 15:33 contradict?

2 Kings 15:30-3 says:

"And Hoshea the son of Elah made a conspiracy against Pekah the son of Remaliah, and smote him, and slew him, and reigned in his stead, in the twentieth year of Jotham the son of Uzziah. And the rest of the acts of Pekah, and all that he did, behold, they are written in the book of the chronicles of the kings of Israel. In the second year of Pekah the son of Remaliah king of Israel began Jotham the son of Uzziah king of Judah to reign. Five and twenty years old was he when he began to reign, and he reigned sixteen years in Jerusalem. And his mother's name was Jerusha, the daughter of Zadok."

There is no contradiction between verse 30 and verse 33.  "Twentieth year of Jotham" does not necessarily mean the twentieth year of his reign as king.  We must not add the words "of his reign" because it does not exist in the text and does not fit the context of verse 33.  The preposition "ל" in the phrase "of Jotham" (ליותם) simply relates "twentieth year" to "Jotham".  Thus "twentieth year of Jotham" simply means the twentieth year after Jotham - namely, the twentieth year after Jotham came on the scene.  Such an interpretation is consistent with the context and does not pose a contradiction.