SOLUTION: If (a)(b^4)(c^3)=1215000, where a, b and c are distinct positive integers greater than 1, what is the greatest possible value of a+b+c.

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Question 1150079: If (a)(b^4)(c^3)=1215000, where a, b and c are distinct positive integers greater than 1, what is the greatest possible value of a+b+c.
Found 3 solutions by ikleyn, MathTherapy, greenestamps:
Answer by ikleyn(52879) About Me  (Show Source):
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The prime decomposition of the number 1215000 is


    1215000 = 3%5E5.5%5E4.2%5E3.


Comparing it with  a%2Ab%5E4%2Ac%5E3, you see that


    either  a= 3^5,  b= 5,    c= 2,      with  a+b+c = 3^5 + 5 + 2   = 250,

    or      a= 3^2,  b= 5,    c = 2*3,   with  a+b+c = 3^2 + 5 + 2*3 =  20,

    or      a = 3,   b= 5*3,  c = 2,     with  a+b+c = 3   + 5*3 + 2 =  20.


So, a+b+c is maximal and equal to 250  at  a= 3^5,  b= 5  and c= 2.    ANSWER

Solved.


Answer by MathTherapy(10556) About Me  (Show Source):
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If (a)(b^4)(c^3)=1215000, where a, b and c are distinct positive integers greater than 1, what is the greatest possible value of a+b+c.
Prime factors of 1,215,000: matrix%281%2C6%2C+2%5E3%2C+%22%2C%22%2C+3%5E5%2C+%22%2C%22%2C+and%2C+5%5E4%29

FACTS: 1) "a" MUST have the largest value

       2) The order of the other 2 prime factors that come after "a" doesn't 

          MATTER, which means that we can either have:  

This then leaves us with: 

Thus, greatest possible value of

Answer by greenestamps(13209) About Me  (Show Source):
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The prime factorization of 1215000 is (2^3)*(3^5)*(5^4). This is to be expressed as (a)(b^4)(c^3), with the sum a+b+c as large as possible.

Since in that form b and c are small numbers, and since we want the sum a+b+c to be as large as possible, we want a to contain as many large factors as possible.

So we want a to contain all 4 factors of 5.

There are only 3 prime factors of 2 in the number, so b can't be 2. So b should be 3, with c = 2.

Then (b^4)(c^3) = (3^4)(2^3); and then a is made up of the remaining factors of the number: (3^1)(5^4).

So

a = 3*5^4 = 3*625 = 1875
b = 3
c = 2

The maximum sum a+b+c is 1875+3+2 = 1880.