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📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 Question: A regular tetrahedron has three of its vertices at $ (0,0,0), (1,0,0), (0,1,0) $. Find the coordinates of the fourth vertex with integer coordinates.

📌 Let the fourth vertex be $ (x, y, z) $. Since the tetrahedron is regular, all edges must have equal length. The distance between any two of the given points is:

📌 Between $ (0,0,0) $ and $ (1,0,0) $: $ 1 $

📌 Between $ (0,0,0) $ and $ (0,1,0) $: $ 1 $

📌 So the edge length is not uniform among the given points. However, in a regular tetrahedron, all six edges must be equal. So we must find a point $ (x, y, z) $ such that its distance to each of the three given points is equal to the edge length of the tetrahedron, and all six edges are equal.

📌 The LCM is obtained by taking the highest powers of all primes appearing:

📌 Question: An angel investor is funding two startups: one receives funding every 7 months, the other every 11 months. If both received funding today, after how many months will both receive funding on the same month again?

📌 Since both 7 and 11 are prime numbers, their LCM is simply their product:

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