Data point: I had 14+ scans in a trip around the world (including an old machine at the entrance to Tianenmen Square) in 2011 with no damage to my film. Fastest were TMY2 & Portra 400. I also left a sacrificial roll of Ektar in my checked luggage for the last 3 flights (including ex-China) with no damage to it.
So 5 passes through standard carry-on scanners is nothing to worry about. I would worry much much more about a fumble-fingered hand-inspection fogging my film.
I vaguely recall someone saying you get a far higher dose merely from being at high altitude for the duration of your typical international flight than you do from the scanners. If you wanted to be paranoid and/or carry very fast film, have a lead bag to put the film in while it's on-board but not while it's being scanned.