Let $m$ (respectively, $n$ ) be the number of $5$ -digit integers obtained by using the digits $1,2,3,4,5$ with repetitions (respectively, without repetitions) such that the sum of any two adjacent digits is odd. Then $\frac{m}{n}$ is equal to

$3,4,5$ without repetitions such that sum of any two adjacent digits is odd.

Sum of two digits are odd if one is even and other is odd.

Even $=2,4$
Odd $=1,3,5$

Case I Digit is repeated.
Two possibilities

(a) odd even odd even odd

(b) even odd even odd even

Case $\mathrm{II}$ Digit is not repeated.
The possibility of arrangement is
odd even odd even odd