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# CAT 2019 Quant Question with Solution 54

Question:
What is the largest positive integer n such that $\frac{n^{2}+7 n+12}{n^{2}-n-12}$ is also a positive integer?

1. 8
2. 12
3. 16
4. 6

$\frac{{{n}^{2}}+7n+12}{{{n}^{2}}-n-12}=\frac{\left( n+3 \right)\left( n+4 \right)}{\left( n-4 \right)\left( n+3 \right)}=\frac{\left( n+4 \right)}{\left( n-4 \right)}$

$\Rightarrow \frac{\left( n+4 \right)}{\left( n-4 \right)}=\frac{\left( n-4+8 \right)}{\left( n-4 \right)}=1+\frac{8}{\left( n-4 \right)}$

The expression is positive integer if $\frac{8}{\left( n-4 \right)}$ is integer.

Or (n-4) must be factor of 8.

For n to be largest, n-4=8

Or n =12

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