How many positive whole-number divisors does 196 have?
A good strategy is to find the prime factorization of 196, and then use the exponent rule for divisors. To find the prime factorization, I can use a factor tree or trial division. I'll start by dividing 196 by 2, the smallest prime number. 196 divided by 2 is 98, so I have 196 = 2 x 98. 98 is not prime, so I need to divide it by another prime. I can divide 98 by 2 again, and get 98 = 2 x 49. 49 is not prime either, so I need to divide it by another prime. I know that 49 is the square of 7, so I have 49 = 7 x 7. 7 is prime, so I can stop here. The prime factorization of 196 is 196 = 2 x 2 x 7 x 7, or 196 = 2^2 x 7^2. Now, to find the number of divisors, I can use the exponent rule: if n = p^a x q^b x r^c, where p, q, and r are distinct primes, then the number of divisors of n is (a + 1)(b + 1)(c + 1). In this case, n = 196, p = 2, a = 2, q = 7, b = 2, and r and c are not relevant. So the number of divisors of 196 is (2 + 1)(2 + 1) = 3 x 3 = 9. 