Question

what is the average of all positive integers that have three digits when written in base $5$, but two digits when written in base $8$? write your answer in base $10$.

Answer 1

The average of all positive integers that have three digits in base 5 but two digits in base 8 is 63 in base 10.

Step-by-step explanation:

Define the range:

Three digits in base 5 means values between 5^2 = 25 and 5^3 = 125 (excluding 25).

Two digits in base 8 means values between 8^1 = 8 and 8^2 = 64 (excluding 8).

Find the intersection:

Convert the base 8 range to base 5:

8 in base 5 is 121 (smallest 3-digit number)

64 in base 5 is 401 (largest 2-digit number)

Therefore, the intersection of these ranges is between 121 and 401 in base 5.

Calculate the average:

To find the average, we need the sum of these integers divided by the total number of integers in the range.

Counting from 121 to 401 in base 5 (inclusive) gives 281 integers.

Unfortunately, calculating the sum directly in base 5 is cumbersome.

Utilize commonalities:

Notice that the range in base 5 has a pattern: it starts and ends with a "1" and has 280 "0"s in between.

This pattern allows us to calculate the sum indirectly:

The sum of all 281 digits in the range is 281 (sum of all "1"s).

Subtract the sum of all "0"s: 281 - (280 * 0) = 281.

Divide the sum by the number of integers: 281 / 281 = 1 (in base 5).

Convert to base 10:

1 in base 5 is equivalent to 5 in base 10 (the base of the digit system).

Therefore, the average of all such integers is 5 in base 10, which is simply 63.

Therefore, the average of these integers in base 10 is 63, highlighting the power of utilizing patterns and commonalities in calculations.

Answer 2

The average of all positive integers that have three digits in base 5 but two digits in base 8 is found by calculating the mean of the smallest and largest numbers within the specified range. The smallest is 25 (base 10) and the largest is 63 (base 10), yielding an average of 44 in base 10.

Step-by-step explanation:

The question asks for the average of all positive integers that are three digits in base 5 but two digits in base 8. To find these numbers, we should understand that the smallest three-digit number in base 5 is 100₅, which is 25 in base 10, and the largest two-digit number in base 8 is 77₈, which is 63 in base 10. The range of numbers we are looking for is between 25 and 63 in base 10. To find the average, we must add up all integers within this range and then divide by the number of integers.

However, without performing the entire calculation, we can rely on the property that the average of a consecutive set of numbers is the mean of the smallest and largest numbers. So, in this case, the average would be (25 + 63) / 2 = 44 in base 10. Since 44 is less than 64 (which is 100₈), it is a two-digit number in base 8 and a three-digit number in base 5 (134₅).

So, the average in base 10 of all positive integers that have three digits in base 5 but two digits when written in base 8 is 44.