Let $n$ be an integer greater than 1. If all digits of $97n$ are odd, find the smallest possible value of $n$.
1. We need to find the smallest integer \( n \) greater than 1 such that all digits of \( 97n \) are odd.
2. First, we note that if \( n \) is even, then \( 97n \) will be even, and thus will have at least one even digit. Therefore, \( n \) must be odd.
3. We start by testing small odd values of \( n \) to see if \( 97n \) has all odd digits.

Let's test \( n = 3 \):
\[ 97 \times 3 = 291 \]
The digits of 291 are 2, 9, and 1. Since 2 is even, \( n = 3 \) does not work.

Let's test \( n = 5 \):
\[ 97 \times 5 = 485 \]
The digits of 485 are 4, 8, and 5. Since 4 and 8 are even, \( n = 5 \) does not work.

Let's test \( n = 7 \):
\[ 97 \times 7 = 679 \]
The digits of 679 are 6, 7, and 9. Since 6 is even, \( n = 7 \) does not work.

Let's test \( n = 9 \):
\[ 97 \times 9 = 873 \]
The digits of 873 are 8, 7, and 3. Since 8 is even, \( n = 9 \) does not work.

Let's test \( n = 11 \):
\[ 97 \times 11 = 1067 \]
The digits of 1067 are 1, 0, 6, and 7. Since 0 and 6 are even, \( n = 11 \) does not work.

Let's test \( n = 13 \):
\[ 97 \times 13 = 1261 \]
The digits of 1261 are 1, 2, 6, and 1. Since 2 and 6 are even, \( n = 13 \) does not work.

Let's test \( n = 15 \):
\[ 97 \times 15 = 1455 \]
The digits of 1455 are 1, 4, 5, and 5. Since 4 is even, \( n = 15 \) does not work.

Let's test \( n = 17 \):
\[ 97 \times 17 = 1649 \]
The digits of 1649 are 1, 6, 4, and 9. Since 6 and 4 are even, \( n = 17 \) does not work.

Let's test \( n = 19 \):
\[ 97 \times 19 = 1843 \]
The digits of 1843 are 1, 8, 4, and 3. Since 8 and 4 are even, \( n = 19 \) does not work.

Let's test \( n = 21 \):
\[ 97 \times 21 = 2037 \]
The digits of 2037 are 2, 0, 3, and 7. Since 2 and 0 are even, \( n = 21 \) does not work.

Let's test \( n = 23 \):
\[ 97 \times 23 = 2231 \]
The digits of 2231 are 2, 2, 3, and 1. Since 2 is even, \( n = 23 \) does not work.

Let's test \( n = 25 \):
\[ 97 \times 25 = 2425 \]
The digits of 2425 are 2, 4, 2, and 5. Since 2 and 4 are even, \( n = 25 \) does not work.

Let's test \( n = 27 \):
\[ 97 \times 27 = 2619 \]
The digits of 2619 are 2, 6, 1, and 9. Since 2 and 6 are even, \( n = 27 \) does not work.

Let's test \( n = 29 \):
\[ 97 \times 29 = 2813 \]
The digits of 2813 are 2, 8, 1, and 3. Since 2 and 8 are even, \( n = 29 \) does not work.

Let's test \( n = 31 \):
\[ 97 \times 31 = 3007 \]
The digits of 3007 are 3, 0, 0, and 7. Since 0 is even, \( n = 31 \) does not work.

Let's test \( n = 33 \):
\[ 97 \times 33 = 3201 \]
The digits of 3201 are 3, 2, 0, and 1. Since 2 and 0 are even, \( n = 33 \) does not work.

Let's test \( n = 35 \):
\[ 97 \times 35 = 3395 \]
The digits of 3395 are 3, 3, 9, and 5. All digits are odd.

Therefore, the smallest possible value of \( n \) such that all digits of \( 97n \) are odd is \( n = 35 \).

The final answer is \( \boxed{35} \).