3. An integer $N<2024$ is divisible by 39 times the sum of its digits. Find all possibilities for $N$.
Solution
As $N$ is a multiple of 39 , it is also a multiple of 3 .
Hence, the sum of its digits is a multiple of 3 .
Hence, $N$ is a multiple of $39 \times 3=117$, and in particular also a multiple of 9 (as $117=9 \times 13$ ). Hence, the sum of its digits is a multiple of 9 .
Hence, $N$ is a multiple of $39 \times 9=351$.
The only multiples of 351 , smaller than 2024, are: $351,702,1053,1404$, and 1755 .
The first four of them each have their digit sum of 9 , hence they are multiples of 39 times the sum of its digits (which is 351 ). In case of 1755, its digit sum is 18. As an odd number, 1755, is not divisible by 18 .
Hence, the only positive integers $N<2024$, divisible by 39 times the sum of its digits, are 351 , 702,1053 , and 1404 .