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 Find the sum of all the natural numbers that are three - digit palindromes when expressed in base 5. Express your answer in base 5. 

 Jul 31, 2023
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The smallest three-digit palindrome in base 5 is 1010, and the largest is 4444. The sum of all the natural numbers that are three-digit palindromes when expressed in base 5 is:

1010 + 1111 + 1212 + ... + 4444

We can factor out a 10 from each term:

10(10 + 11 + 12 + ... + 44)

The sum 10 + 11 + 12 + ... + 44 is a geometric series with first term 10 and common ratio 11/10. The sum of a geometric series is equal to the first term divided by 1 - the common ratio. In this case, the sum is equal to:

10 / (1 - 11/10) = 100

Therefore, the sum of all the natural numbers that are three-digit palindromes when expressed in base 5 is:

10(100) = 1000

In base 5, this is equal to 10000.

 Jul 31, 2023

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