One ampere-hour is exactly 3,600 coulombs, because a coulomb is one ampere-second and an hour is 3,600 seconds. The coulomb is the SI unit; the ampere-hour is what industry actually uses.
About the units
The coulomb is the SI unit of electric charge, defined as the charge transported by a current of one ampere in one second. Since the 2019 SI revision it is tied to the elementary charge, which was fixed at exactly 1.602176634 × 10⁻¹⁹ coulombs — meaning a coulomb is exactly 6.241509074 × 10¹⁸ elementary charges. The ampere-hour is not an SI unit but is universal in battery work, because battery capacity is naturally described by how long a given current can be sustained.
The exact factor
The factor is exactly 3,600. A 2,500 mAh AA cell therefore holds 9,000 coulombs, and a 100 Ah car battery holds 360,000. Those figures are rarely quoted because the coulomb is inconveniently small for the application, but they matter in electrochemistry: the Faraday constant, 96,485 coulombs per mole, links charge to the amount of substance deposited or dissolved, and that is how electroplating thickness and battery capacity are calculated from first principles.
Where you meet this conversion
Electrochemistry and physics rather than everyday battery work. The laws of electrolysis relate charge in coulombs to moles of substance, so plating, anodising and electrowinning calculations all use the coulomb. Capacitor charge, static electricity and particle beam current are quoted in coulombs or their prefixed forms. Battery research reports specific capacity in ampere-hours per kilogram while the underlying electrochemistry works in coulombs per mole, and the conversion bridges the two.