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.