Rocks & Boxes
In a long hallway are 100 boxes numbered 1 through 100 (in that order). Three people walk through the hallway dropping rocks into boxes as follows. Adam drops a rock in Box 1 and every other box after that (3, 5, 7, etc). Beth drops a rock in Box 1 and every third box after that (4, 7, etc). Carl drops a box in Box 1 and every fifth box after that (6, 11, etc). When all rocks have been dropped how many boxes have exactly two rocks?
The answer is 22.
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1 solution
Since they are all in 1, I'll label Box 1 "Box #0", and change Box n to "Box #(n-1)", so the last box is Box#99.
Adam drops the rocks in Box #2n, Beth drops the rocks in Box#3n, Carl drops the rocks in Box #5n.
So the boxes that has exactly two rocks is equal to
N ( 6 n ) + N ( 1 0 n ) + N ( 1 5 n ) − 3 N ( 3 0 n ) = 1 6 + 9 + 6 − 3 × 3 = 2 2