You can put this solution on YOUR website! The next number is 0. Every number thereafter is also 0. The pattern is that the next number is the product of the digits of the previous number.
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77 (7*7=49) -> 49 (4*9=36) -> 36 (3*6=18) -> 18 (1*8=8) -> 08 (0*8=0) -> 0 (0*0) -> 0