How many ordered triples (x,y,z) of positive integers satisfy lcm(x,y) = 72, lcm(x,z) = 600 and lcm(y,z)=900$?
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We need to find ordered triples of positive integers x, y, z satisfying three LCM conditions. First, let's find the prime factorizations of 72, 600, and 900. We have 72 equals 2 cubed times 3 squared, 600 equals 2 cubed times 3 times 5 squared, and 900 equals 2 squared times 3 squared times 5 squared.
Now let's set up our variables. We express x, y, and z in terms of their prime factorizations using exponents a, b, and c. Since LCM takes the maximum exponent for each prime, our three conditions translate to systems of max equations. For each prime factor, we get three constraints on the exponents.
Let's solve for prime factor 2. From the constraint that max of b₁ and c₁ equals 2, both b₁ and c₁ must be at most 2. This forces a₁ to equal 3, since otherwise the other max conditions couldn't be satisfied. We need pairs where the maximum is exactly 2, giving us 5 valid solutions.
For prime factor 3, the constraints force b₂ to equal 2, and we need max of a₂ and c₂ to equal 1, giving us 3 solutions. For prime factor 5, the first constraint forces both a₃ and b₃ to be 0, and the other constraints force c₃ to be 2, giving only 1 solution. The total number of ordered triples is 5 times 3 times 1, which equals 15.
To summarize our solution: we found 5 solutions for the exponents of prime 2, 3 solutions for prime 3, and 1 solution for prime 5. Since the choices for each prime are independent, we multiply these together to get 5 times 3 times 1 equals 15. Therefore, there are exactly 15 ordered triples of positive integers satisfying the given LCM conditions.