How many tanks in a modern us army armoured division?

The United States Army is currently reorganizing around the Brigade as the largest permanent flexible unit of combat force, and the division is much less fixed. Indeed, the idea is that divisions will be amalgamations of standing Brigades (usually 2-4) plus additional support services. In the interim, US Divisions are being redesigned to be able to deploy independent brigades.

A rough estimate of a current full armored division is the composition of the US 1st Armored Division, which is comprised of 4 Brigade Combat Teams, totaling 4 tank battalions and 3 armored cavalry battalions. A tank battalion has 3 companies, each of roughly 14 tanks. Armored Cavalry battalions generally have half as many tanks. So, a good estimate would be that the 1st Armored Division has about 250 tanks.

Historically, it is very hard to answer this question, as it depends heavily on which period of time you are talking about - the size and composition of US armored divisions has varied considerably over the past 70 years.