Math Problem Statement
Suppose a, b, c, and d are positive integers with a < b < c < d. Consider the integer (b − a)(c − a)(d − a)(c − b)(d − b)(d − c) obtained by multiplying together the 6 possible positive differences between the four integers. Determine the largest integer n for which it is guaranteed that this product is a multiple of n. Hint: you can substitute x for b - a, y for c - b, and z for d - c, and come up with a new equation: xyz(x + y)(x + y + z)(z + y).
Solution
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Math Problem Analysis
Mathematical Concepts
Number Theory
Prime Factorization
Modular Arithmetic
Formulas
-
Theorems
Pigeonhole Principle
Suitable Grade Level
Grades 10-12
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