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

38

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.