There are 8,181 five-digit multiples of 11.

How to see it step by step

All five-digit numbers run from 10,000 to 99,999.

  1. Find the first five-digit multiple of 11 Divide 10,000 by 11:

10,000÷11≈909.something10{,}000\div 11\approx 909.\text{something}10,000÷11≈909.something

So the first integer nnn with 11n≥10,00011n\ge 10{,}00011n≥10,000 is n=910n=910n=910, and

11×910=10,01011\times 910=10{,}01011×910=10,010

So 10,010 is the smallest five-digit multiple of 11.

  1. Find the last five-digit multiple of 11 Divide 99,999 by 11:

99,999÷11≈9,090.something99{,}999\div 11\approx 9{,}090.\text{something}99,999÷11≈9,090.something

So the largest integer nnn with 11n≤99,99911n\le 99{,}99911n≤99,999 is n=9,090n=9{,}090n=9,090, and

11×9,090=99,99011\times 9{,}090=99{,}99011×9,090=99,990

So 99,990 is the largest five-digit multiple of 11.

  1. Count how many multiples there are Now we are counting all integers nnn such that

910≤n≤9,090910\le n\le 9{,}090910≤n≤9,090

The count of integers from 910 to 9,090 inclusive is

9,090−910+1=8,1819{,}090-910+1=8{,}1819,090−910+1=8,181

So, there are 8,181 five-digit numbers that are multiples of 11.