Can you play an xbox 360 game on xbox live with 2 different consoles (at different times)?

my roomate this year for college brought his xbox 360, so naturally i left mine at home- no need to lug it here. well i am going to buy halo reach this tuesday and im worried that if i play it on his console this week i won't be able to play the same disc on my own console at home this weekend. does any1 know if you can do this? also if i install it to his hardrive, is that better or worse, or does it matter. thanks in advance.