Question

For a nonnegative integer $n$, let $r_9(n)$ stand for the remainder left when $n$ is divided by $9.$ For example, $r_9(25)=7.$ What is the $22^{\text{nd}}$ entry in an ordered list of all nonnegative integers $n$ that satisfy $r_9(5n)\le 4~?$(Note that the first entry in this list is $0$.)

Answer 1

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Answer 2

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.