+90 How many positive integers are factors of either 144 or 196 but not both?
THE ANSWER IS NOT 21.
The prime factorization of 144 is $2^4 \cdot 3^2$ and the prime factorization of 196 is $2^2 \cdot 7^2$. To count the number of factors of each number, we add 1 to each exponent in the prime factorization and then multiply. Thus, 144 has $(4+1)(2+1)=15$ factors and 196 has $(2+1)(2+1)=9$ factors.
To count the number of factors of either 144 or 196 but not both, we need to count the factors of 144 that are not factors of 196, the factors of 196 that are not factors of 144, and the factors that are common to both.
The prime factors of 144 that are not in 196 are $2^2$ and $3^2$. Thus, there are $(2+1)(2+1)-1=8$ factors of 144 that are not factors of 196.
The prime factors of 196 that are not in 144 are $7^2$. Thus, there is $1$ factor of 196 that is not a factor of 144.
Finally, the factors that are common to both 144 and 196 are those that have prime factors of $2^2$ and $7^2$, of which there is $1$.
Therefore, the total number of positive integers that are factors of either 144 or 196 but not both is $8+1=\boxed{9}$.
