a) Find all integers that can divide both the numerator and denominator of the ratio $\frac{5m + 6}{8m + 7}$ for an integer $m$.
b) Let $a, b, c, d, m$ be integers. Prove that if the numerator and denominator of the ratio $\frac{am + b}{cm+ d}$ are both divisible by $k$, then so is $ad - bc$.
### Part (a)
1. We need to find all integers \( x \) that can divide both the numerator \( 5m + 6 \) and the denominator \( 8m + 7 \) for any integer \( m \).
2. If \( x \) divides both \( 5m + 6 \) and \( 8m + 7 \), then \( x \) must also divide any linear combination of these two expressions. Consider the following:
   \[
   x \mid 5m + 6 \quad \text{and} \quad x \mid 8m + 7
   \]
3. Multiply the first equation by 8 and the second by 5:
   \[
   x \mid 8(5m + 6) = 40m + 48 \quad \text{and} \quad x \mid 5(8m + 7) = 40m + 35
   \]
4. Subtract the second equation from the first:
   \[
   x \mid (40m + 48) - (40m + 35) = 48 - 35 = 13
   \]
5. Therefore, \( x \) must divide 13. The divisors of 13 are \( \pm 1 \) and \( \pm 13 \).

Conclusion:
\[
\boxed{\pm 1, \pm 13}
\]

### Part (b)
1. We need to prove that if \( k \) divides both the numerator \( am + b \) and the denominator \( cm + d \) of the ratio \( \frac{am + b}{cm + d} \), then \( k \) also divides \( ad - bc \).
2. Given:
   \[
   k \mid am + b \quad \text{and} \quad k \mid cm + d
   \]
3. This implies:
   \[
   am + b = kq_1 \quad \text{for some integer } q_1
   \]
   \[
   cm + d = kq_2 \quad \text{for some integer } q_2
   \]
4. Multiply the first equation by \( c \) and the second by \( a \):
   \[
   c(am + b) = c(kq_1) \implies cam + cb = kcq_1
   \]
   \[
   a(cm + d) = a(kq_2) \implies acm + ad = kaq_2
   \]
5. Subtract the second equation from the first:
   \[
   cam + cb - (acm + ad) = kcq_1 - kaq_2
   \]
   \[
   cb - ad = k(cq_1 - aq_2)
   \]
6. Therefore, \( k \) divides \( cb - ad \), which can be rewritten as \( ad - bc \) by changing the sign.

Conclusion:
\[
\boxed{\text{None}}
\]