Weekend Math Problems: Can 2024 Be Written as a⁵ + b³? And Divisibility by 9

Last week’s problem was straightforward — but this week you’ll need to think a little harder!

Weekend Math Problems: Two Challenges to Test Your Skills

Problem 1
Can the number 2024 be expressed as a sum a⁵ + b³, where a and b are natural numbers?

Problem 2
A certain number contains 10 ones, 100 twos, 1000 threes, …, 10⁹ nines, arranged in some order. Every second, the last digit is erased. Is it true that at some point during this process the resulting number will be divisible by 9?

Try to solve them yourself before reading the solutions below!

Solutions to This Week’s Daily Math Challenge

Problem 1 — Solution:

Yes, it is possible. Notice that:

2024 = 1024 + 1000 = 4⁵ + 10³

Problem 2 — Solution:

Yes, it is true.

Why Divisibility by 9 Is Guaranteed: A Step-by-Step Breakdown

The remainder of a number when divided by 9 equals the remainder of the sum of its digits divided by 9. With that in mind, we can mentally cross out all the nines, since they don’t affect the remainder.

Our number contains 10⁸ eights, and the count of all other digits is 1 + 10 + 100 + … = 111,111,110.

Now split those 111,111,110 digits into 11,111,111 blocks of 10. At least one of these blocks consists entirely of eights. Suppose that after erasing the first digit from such a block, the remaining digit sum equals x. Then as we erase the next eight eights one by one, the running digit sums become x − 8, x − 16, …, x − 64.

This gives us 9 numbers whose remainders modulo 9 are all distinct — meaning they cover every possible remainder class from 0 to 8. Therefore, at least one of them is divisible by 9.

The Answer

Since among those nine intermediate values there must be at least one that is a multiple of 9, it is guaranteed that at some point during the process the resulting number will be divisible by 9.