CRC Procedure - Checking Efficiently

CRC Procedure - Checking Efficiently

Let us get an m bit-message where the last n bits are the CRC bits. As far
as I know, in order to check if it is received correctly or not, we should
XOR all m bits with the polynomial of the specific CRC algorithm. If the
result is all-zeros, we can say there are no errors.
Here are my questions:
1) What about calculating the n CRC bits using the first (m-n) bits and
then compare it to the last n bits of the received message? This way we
can say there are no errors if the received and calculated n bits are
equal. Is this approach true?
2) If it is true, which is more efficient?