Maths Olympiad Prep

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Geometry Difficulty 4.7 AIME Find the answer

9. Several rectangular prisms with edge lengths of 2,3,52, 3, 5 are combined in the same direction to form a cube with an edge length of 90. The number of small rectangular prisms that a diagonal of the cube passes through is:

Pick one

Solution

9. B.

Since the least common multiple of 2,3,52,3,5 is 30, and the number of small rectangular prisms with edge lengths of 2,3,52,3,5 that a diagonal of a cube with edge length 30 passes through is [302]+[303]+[305][302×3][302×5][303×5]+[302×3×5]=22\left[\frac{30}{2}\right]+\left[\frac{30}{3}\right]+\left[\frac{30}{5}\right]-\left[\frac{30}{2 \times 3}\right]-\left[\frac{30}{2 \times 5}\right]-\left[\frac{30}{3 \times 5}\right]+\left[\frac{30}{2 \times 3 \times 5}\right]=22, therefore, the number of small rectangular prisms that a diagonal of a cube with edge length 90 passes through should be 3×22=663 \times 22=66.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.