Not only is 288 a multiple of 36, it is also a multiple of 12. The GR"A thus told his Talmidim to count 24 times to 12, eight times to 36. Moreover, we have long known the count of 12 years as the Chesed count.
So the GR"A count does not start in 5500, but in 5501. We are not to count every 250/7 years, but every 36 years. The holy spirit of the GR"A told us that the era of Moshiach ben Yosef is 24 times 12 years, 24 years of Chesed, but he could not say it this way, and the wise will understand.