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Find the number of ordered triples (a, b, c) of positive integers…
Question
Find the number of ordered triples (a, b, c) of positive integers such that a · b · c = 10800. (Note 10800 = 2^4 · 3^3 · 5^2.)
✓ Verified answer: 900checked by our engine — not a guess
Step-by-step solution
Write a = 2^x1·3^y1·5^z1 and similarly for b, c.
The condition abc = 2^4·3^3·5^2 splits independently across primes: we need x1+x2+x3 = 4, y1+y2+y3 = 3, z1+z2+z3 = 2 with nonnegative integers.
The number of solutions of e1+e2+e3 = E is C(E+2, 2).
So counts are C(6,2)=15, C(5,2)=10, C(4,2)=6.
Total = 15·10·6 = 900.
Final answer900
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