#
Problem
of the Week

Problem
D and Solution

Missing
the Fives II

## Problem

Bobbi lists the positive integers, in order, excluding all multiples
of 5. Her resulting list is \[1, 2, 3, 4, 6,
7, 8, 9, 11, 12, 13, 14, 16, 17, \ldots\] If the \(n\)th integer in Bobbi’s list is \(2023\), what is the value of \(n\)?

## Solution

**Solution 1**

Note that \(2023\) is two integers
before \(2025\), which is a multiple of
\(5\).

Beginning at \(1\), \(2025\) is the \(405\)th multiple of \(5\), since \(\frac{2025}{5} = 405\). That is, the
integers from \(1\) to \(2025\) contain \(405\) groups of \(5\) integers.

Each of these \(405\) groups contain
one integer that is a multiple of \(5\), and so Bobbi leaves out \(406\) integers (including \(2024\)) in the list of all integers from
\(1\) to \(2025\). If the \(n\)th integer in Bobbi’s list is \(2023\), then \(n
= 2025 - 406 = 1619\).

**Solution 2**

Note that \(2023\) is two integers
before \(2025\), which is a multiple of
\(5\).

Beginning at \(1\), \(2025\) is the \(405\)th multiple of \(5\), since \(\frac{2025}{5} = 405\). That is, the
integers from \(1\) to \(2025\) contain \(405\) groups of \(5\) integers.

In Bobbi’s list of integers, she leaves out the integers that are
multiples of \(5\), and so in every
group of five integers, Bobbi lists four of these integers. However, she
also does not list \(2024\). Thus, if
the \(n\)th integer in Bobbi’s list is
\(2023\), then \(n = 405 \times 4 - 1= 1619\).