Question 1171174: If a,b, and c are positive integers, what is the sum of a+b+c if:
a^3 x b^3 x c = 454 425 283
a^3 x b x c^3 = 248 258 803
a x b^3 x c^3 = 777 094 123
Answer by math_tutor2020(3817)   (Show Source):
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Find the prime factorization of each right hand side
454,425,283 = 13^3*17*23^3
248,258,803 = 13^3*17^3*23
777,094,123 = 13*17^3*23^3
Each prime factorization involves 13, 17, and 23.
So we'll have {a,b,c} take on values from {13,17,23}
Rearranging terms a bit, we get,
a^3*b^3*c = 454,425,283
a^3*b^3*c = 13^3*17*23^3
a^3*b^3*c = 13^3*23^3*17
for the first equation. Which suggests that a = 13, b = 23, c = 17
and doing the same for the second equation leads to
a^3*b*c^3 = 248,258,803
a^3*b*c^3 = 13^3*17^3*23
a^3*b*c^3 = 13^3*23*17^3
which also suggests a = 13, b = 23, c = 17
Finally the third equation
a*b^3*c^3 = 777,094,123
a*b^3*c^3 = 13*17^3*23^3
which also leads to a = 13, b = 17, c = 23
Therefore,
a+b+c = 13+17+23 = 53
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Answers:
a = 13
b = 17
c = 23
a+b+c = 53
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