How many of the following numbers are factors of 34 or multiples of 7?

1, 2, 3, 4, 8, 14, 17, 29, 56, 91
We can check each number one by one.

1: 1 is a factor of 34 since $1\cdot34=34$ .

2: 2 is a factor of 34 since $2\cdot17=34$ .

3: 3 is not a factor of 34 since there is no number that can be multiplied by 3 to get 34. ( $34\div3$ gives a quotient of 11 and a  remainder of 1.) There is also no number that can be multiplied by 7 to get 3 ( $3\div7$ gives a quotient of 0 and a remainder 3.)

4: 4 is not a factor of 34 since there is no number that can be multiplied by 4 to get 34.  ($34 \div 4$ gives a quotient of 8 and a remainder of 2.)  There is also no number that can be multiplied by 7 to get 4.  ($4 \div 7$ gives a quotient of 0 and a remainder of 4.)

8:  8 is not a factor of 34 since there is no number that can multiply it to get 34 ( $34\div8$ gives a quotient of 4 and a  remainder of 2) and is not a multiple of 7 since there is no number that can multiply 7 to get 8 ( $8\div7$ gives a quotient of 1 and a remainder 1).

14: 14 is a multiple of 7 since $7\cdot2=14$ .

17: 17 is a factor of 34 since $17\cdot2=34$ .

29: 29 is not a factor of 34, since there is no number that can multiply it to get 34 ( $34\div29$ gives a quotient of 1 and a  remainder of 5) and is not a multiple of 7 since there is no number that can multiply 7 to get 29 ( $29\div7$ gives a quotient of 4 and a remainder 1).

56: 56 is a multiple of 7 since $7\cdot8=56$ .

91: 91 is a multiple of 7 since $7\cdot13=91$ .

So, $\boxed{6}$ of the 10 numbers are factors of 34 or multiples of 7.