document.write( "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. \n" ); document.write( "
Algebra.Com's Answer #771469 by greenestamps(13200)\"\" \"About 
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\n" ); document.write( "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.

\n" ); document.write( "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.

\n" ); document.write( "So we want a to contain all 4 factors of 5.

\n" ); document.write( "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.

\n" ); document.write( "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).

\n" ); document.write( "So

\n" ); document.write( "a = 3*5^4 = 3*625 = 1875
\n" ); document.write( "b = 3
\n" ); document.write( "c = 2

\n" ); document.write( "The maximum sum a+b+c is 1875+3+2 = 1880.

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