### Instruction: What is the greatest 3-digit base 8 positive integer that is divisible by 5?  (Express your answer in base 8.).### Response: I know that base 8 means that each digit represents a power of 8, so the largest 3-digit base 8 number is 777, which means 7 * 8^2 + 7 * 8 + 7 = 511 in base 10.
I want to find the largest multiple of 5 that is less than or equal to 511, since 5 is not a factor of 8 and therefore cannot be a digit in base 8.
I can use long division or a calculator to find that 511 / 5 = 102 with a remainder of 1, so the largest multiple of 5 that is less than or equal to 511 is 102 * 5 = 510.
Now I need to convert 510 from base 10 to base 8.
I can use repeated division by 8 to find the base 8 digits from right to left.
510 / 8 = 63 with a remainder of 6, so the rightmost digit is 6.
63 / 8 = 7 with a remainder of 7, so the next digit is 7.
7 / 8 = 0 with a remainder of 7, so the leftmost digit is 7.
Therefore, the largest 3-digit base 8 positive integer that is divisible by 5 is 776.