Question

After Smaug the dragon was slain, Bilbo and Thorin’s band of 13 Dwarves (for 14 members total) arranged to split the treasure they found 14 equal ways. However, when they tried to divide the golden coins 14 equal ways, they had 9 extra coins. Although dwarves are known to be greedy, they did not fight over who should get the extra coins, since the Battle of Five Armies killed the dwarf leader Thorin. They then tried to divide the coins 13 equal ways, finding that there were 5 extra coins. Reminded that two other dwarves were killed during this battle, they were able to divide the coins in 11 equal ways to everyone’s relief. Assuming that there were at least 5000 golden coins in Smaug’s cache, what’s the least number of golden coins that Bilbo and company had taken?

Answer 1

The least number of coins that Bilbo and company had taken is 5049 coins.

Explanation:

First condition

The required total number of coins is at least 5000.

Second condition

When the number was divided by 14, the remainder was 9.

Third condition

When the number was divided by 13, the remainder was 5.

Fourth condition

But the number is divisible by 11.

Using trial and error method.

Combining the first and fourth conditions;

The least number above 5000 that is divisible by 11 is 5005.

Then, the second condition,

(5005/14) = 357 remainder 7, (remainder should be 9)

Can't be the answer.

5005 + 11 = 5016 (still divisible by 11)

(5016/14) = 358 remainder 4. (Remainder should be 9)

Can't be the answer either.

5016+11 = 5027 (still divisible by 11)

(5027/14) = 359 remainder 1. (Remainder should be 9)

Can't be the answer too.

5027+11 = 5038 (still divisible by 11)

(5038/14) = 359 remainder 12. (Remainder should be 9)

Can't be the answer too.

5038+11 = 5049 (still divisible by 11)

(5049/14) = 360 remainder 9. (Remainder is 9. This number satisfies three of the conditions; could be the answer)

(5049/13) = 388 remainder 5 (satisfies all the four conditions; this is the answer!)

Therefore, the least number of coins that Bilbo and company had taken is 5049 coins.

Hope this Helps!!