jmc

First prime factorize $196=2^2\cdot 7^2$. The prime factorization of any divisor of 196 cannot include any primes other than 2 and 7. We are free to choose either 0, 1, or 2 as the exponent of 2 in the prime factorization of a divisor of 196. Similarly, we may choose 0, 1, or 2 as the exponent of 7. In total, there are $3\times 3=9$ possibilities for the prime factorization of a divisor of 196. Distinct prime factorizations correspond to distinct integers, so there are $\boxed{9}$ divisors of 196.