Final answer:
To find the number of positive even integers less than 10000 that have no 7s as a digit, we consider the place value of each digit. The total number of positive even integers less than 10000 with no 7s as a digit is 4500.
Step-by-step explanation:
To find the number of positive even integers less than 10000 that have no 7s as a digit, we need to consider the place value of each digit. Since we are looking for even numbers, the units digit can only be 0, 2, 4, 6, or 8. The other digits can be any number from 0 to 9, except 7.
For the units digit, we have 5 choices. For the thousands digit, we have 9 choices (0-9 except 7). For the hundreds and tens digits, we have 10 choices each. Therefore, the total number of positive even integers less than 10000 with no 7s as a digit is 5 x 9 x 10 x 10 = 4500.