Answer:
38
Step-by-step explanation:
The condition r_9(5n)<= 4 can also be stated as 5n==0, 1, 2, 3, or 4 (mod 9).
We can then restate that condition again by multiplying both sides by 2: 10n == 0, 2, 4, 6, or 8 (mod 9). This step is reversible (since 2 has an inverse modulo 9). Thus, it neither creates nor removes solutions. Moreover, the left side reduces to n modulo 9, giving us the precise solution set n == 0, 2, 4, 6, or 8 (mod 9). We wish to determine the 22nd nonnegative integer in this solution set. The first few solutions follow this pattern:
0 2 4 6 8
9 11 13 15 17
18 20 22 24 26
27 29 31 33 35
36 38 ...
The 22nd solution is 38.
This was written by an AoPS Staff member.
