For the first one

Given

\(5^2\) and \(3^3\) are factors of \(n\times 2^5\times6^2\times7^3\)

Need to find the smallest value of n.

As \(5^2\) and \(3^3\) are factors of \(n\times 2^5\times6^2\times7^3\),

\(n\times 2^5\times6^2\times7^3\) should be either the Least common Multiple of the two or a multiple of the LCM itself. i.e. dividing \(n\times 2^5\times6^2\times7^3\) by the LCM should result into an integer.

Lets find the LCM of the two:

\(5^2=5\times5\)

\(3^3=3\times3\times3\)

\(\text{LCM}=5\times5\times3\times3\times3=5^2\times3^3\)

Now

\(\frac{n\times 2^5\times6^2\times7^3}{5^2\times3^3}\)

should be an integer

\(=\frac{n\times 2^5\times2^2\times3^2\times7^3}{5^2\times3^3}\)

Reducing it further

\(= \frac{n\times 2^5\times2^2\times7^3}{5^2\times3}\)

for the fraction to be an integer n should be divisible by \({5^2\times3}\). Ans the smallest value n can have is \({5^2\times3}\).

As

\(\frac{5^2\times3}{5^2\times3}=1\)

\({5^2\times3}=75\)

Hence the answer is D.

Also here is a good overview of the basics of factors.

http://www.math.com/school/subject1/les ... 3L1GL.html--== Message from GMAT Club Team ==--

This is not a quality discussion. It has been retired. If you would like to discuss this question please re-post it in the respective forum. Thank you!

To review the GMAT Club's Forums Posting Guidelines, please follow these links:

Quantitative |

Verbal Please note - we may remove posts that do not follow our posting guidelines. Thank you.