Question

After months of planning, 13 robbers break into Fort Knox and steal a number of gold bars. They escape with the goods safely and reach a hideout. They decide to divide the gold bars evenly between them and then split up.

The leader of the group begins to hand out the bars one by one. But when he gets to the last few bars, he realises there are 3 extra bars. The robbers begin to argue over who gets more bars. The argument gets heated, and one of the robbers pulls out a knife. He is immediately thrown out by the leader of the group and forced to return his gold bars.

Everyone calms down, and the leader resumes the distribution, handing one bar at a time to each robber. As the pile of bars is almost depleted, the leader realizes that there are 5 extra bars.

Once again, the robbers start arguing over the extra bars. Tempers run high, another robber pulls out his gun and is subsequently kicked out by the leader, and forced to return his bars of gold.

Then the leader continues handing out the gold bars. This time, there are no extra bars and each robber gets an equal amount of gold. They each pack their gold, and leave the hideout.

Given that there were less than 700 gold bars, how many bars did the robbers have to divide among themselves?

Answer 1

Suppose, robbers had to divide x gold bars among themselves.

We observe 3 scenarios consecutively.

Scenario 1-

Stolen x gold bars were to be divided among 13 robbers. There were 3 extra bars after dividing the gold bars evenly. Thus possible values of x are as follows.

Scenario 2-

Stolen x gold bars were to be divided among 12 robbers. There were 5 extra bars after dividing the gold bars evenly. Thus possible values of x are as follows.

We highlighted those numbers which are feasible both in scenario 1 and scenario 2.

Scenario 3-

Stolen x gold bars were to be divided among 11 robbers. There were no extra bars after dividing the gold bars evenly. Thus possible values of x are as follows.

We highlighted those numbers which are feasible in scenario 1, scenario 2 and scenario 3.

Hence, the robbers had to divide 341 gold bars among themselves.