sorry for the above typos it should read (7^8) + 1.

It seems to me that the answer should be 7^{8 – 1 based on this, “He takes out his 1/7 share then divides the remaining coconuts into 7 piles, and reburies the pile.” This may be a typo and supposed to say 6 but as is, The last guy wouldn’t be able to make 7 even piles after he removed his share unless they started with 7}8 – 1.

This continues until the 7th person, after which there are 6^7/77 of the original coconuts left buried in the original hole. This number has to be an integer, so the original number buried has to be divisible by 7^{7. The smallest such number is 7}7 itself. One coconut was thrown away, so there were originally 7^7 + 1 coconuts.

No, the answer is right. After the first person redistributes the coconuts, the total number of coconuts left is 6/7 of the original amount. After the second person the total is down to (6/7)*(6/7) = 6^2/72.