Euclids + Euclids Extended for 1776 and 1492

 1: 1776 = 1 * 1492 + 284
 2: 1492 = 5 * 284 + 72
 3: 284 = 3 * 72 + 68
 4: 72 = 1 * 68 + 4
 5: 68 = 17 * 4 + 0
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 6: 4 = 72 - 1(68)           # start from line 4, solved for the gcd
 7: 4 = 72 - 68
 8: 4 = 72 - (284-3(72))     # substitute from line 3, solved for 68
 9: 4 = (1492 - 5(284)) - (284-3((1492 - 5(284))))     # substitute from line 2, solved for 72
10: 4 = (1492 - 5((1776 - 1492))) - ((1776 - 1492)-3((1492 - 5((1776 - 1492)))))     # substitute from line 1, solved for 284
11: a = 1776 # this just makes it easier to follow, we'll replace
12: b = 1492 # our inputs with a and b:
13: 4 = (b - 5((a - b))) - ((a - b)-3((b - 5((a - b)))))     # replaced our inputs with a and b
14: 4 = b - 5a + 5b -a + b +3b -15a +15b) # distribute/multiply constants
15: 4 = (-5-1-15)a + (1+5+1+3+15)b # combine like terms
16: 4 = -21a + 25b # waahlaa!