document.write( "Question 409629: Given 60=2^2 x 3 x 5 and 1050=2 x 3x 5^2 x 7, find
\n" ); document.write( "(a) the smallest integer m such that 60m is a perfect cube
\n" ); document.write( "(b) the smallest positive integer value of n for which 60n is a multiple of 1050. \n" ); document.write( "

Algebra.Com's Answer #288370 by richard1234(7193) $\"\"$ $\"About$

You can put this solution on YOUR website!
Since $\"60+=+2%5E2%2A3%2A5\"$ we can find the smallest (positive) integer such that the prime factors of 60 all have exponents of multiples of 3. Since the exponents of 2, 3, and 5 are 2, 1, 1 respectively, then m can have 2, 3, 5 exponents of 1, 2, 2 respectively. Therefore $\"m+=+2%2A%283%5E2%29%2A%285%5E2%29+=+450\"$ .\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "We want $\"60n%2F1050\"$ to be an integer. The fraction reduces to $\"2n%2F35\"$ , hence n = 35 is the smallest integer value, as 2 and 35 are relatively prime. \n" ); document.write( "

\n" );