## Perfect Cube 735

What is the smallest positive integer by which you can multiply 735 so that the product is a perfect cube?
Source: mathcontest.olemiss.edu 4/14/2008

SOLUTION
Example of a perfect cube: $216=27\cdot 8=3^3\cdot 2^3$.

$735=3\cdot 5\cdot 7^2$.
To make 735 a perfect cube all we have to do is multiply it by $3^2\cdot 5^2\cdot 7=1575$ so that we end up with $3\cdot 3^2\cdot 5\cdot 5^2\cdot 7^2\cdot 7=3^3\cdot 5^3\cdot 7^3$ a perfect cube.

Check
$735\cdot 1575=1157625$
$\sqrt[3]{1157625}=105$

Retired high school math teacher.
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### 2 Responses to Perfect Cube 735

1. vinod kumar says:

sir, please provide me solution of
Question:-find the least number which is added to 200 to make it a perfect cube.

• mvtrinh says:

$5^3=125=200+\left (-75\right )$
$6^3=216=200+16$
$7^3=343=200+143$
$8^3=512=200+312$
Therefore, 16 is the smallest positive integer that is added to 200 to make it a perfect cube.