#
Problem
of the Week

Problem
C and Solution

Missing
the Fives I

## Problem

Bobbi lists the positive integers, in order, excluding all multiples
of \(5\). Her resulting list is \[1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17,
\ldots\] How many integers has Bobbi listed just before she
leaves out the \(2023\)rd multiple of
\(5\)?

## Solution

**Solution 1**

In the list of integers beginning at \(1\), the \(2023\)th multiple of \(5\) is \(2023
\times 5 = 10\,115\). Thus, Bobbi has listed each of the integers
from \(1\) to \(10\,114\) with the exception of the
positive multiples of \(5\) less than
\(10\,115\). Since \(10\,115\) is the \(2023\)rd multiple of \(5\), Bobbi will not write \(2022\) multiples of \(5\).

Therefore, just before Bobbi leaves out the \(2023\)rd multiple of \(5\), she has listed \(10\,114 - 2022 = 8092\) integers.

**Solution 2**

Beginning at \(1\), each group of
five integers has one integer that is a multiple of \(5\). For example, the first group of five
integers, \(1\), \(2\), \(3\), \(4\), \(5\), has one multiple of \(5\) (namely \(5\)), and the second group of five
integers, \(6\), \(7\), \(8\), \(9\), \(10\), has one multiple of \(5\) (namely \(10\)). 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. Thus, just before Bobbi
leaves out the \(2023\)rd multiple of
\(5\), there were \(2023\) of these groups. Therefore, she has
listed \(2023 \times 4 = 8092\)
integers.