Diophantine Equations CMT : 2011 - 2012
نویسنده
چکیده
Solution 1.1.1. The prime factorization of 720 is 2 · 3 · 5. If we look only at the prime factors of x, y, z, we see that the exponents of 2 in the factorizations of x, y, z must sum to 4, the exponents of 3 must sum to 2, and the exponents of 5 must sum to 0. The number of possible triples (a, b, c) that correspond to the exponents of 2 in the prime factorizations of x, y, z is simply the number of number of nonnegative solutions to a + b + c = 4, which is 15. Similarly, we find that the number of triples corresponding to the exponent of 3 is 6, and the number of triples corresponding to the exponent of 5 is 3. Because the exponents of 2, 3, and 5 are all independent of one another, the number of pairs (x, y, z) is simply 15 · 6 · 3 = 270.
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