Question

How many three-digit numbers are divisible by 11? Also find the middle term.

Answer 1

Hint: We first find the least and highest three-digit numbers which are divisible by 11. The number of terms in between gives the solution. We also find the multipliers of the numbers and find the middle point of that series.

Complete step by step answer:
The three-digit numbers are 100 to 999.
To find the three-digit numbers which are divisible by 11, we first find the least and highest three-digit numbers which are divisible by 11.
$11\stackrel{9}{\stackrel{―}{)\begin{array}{rl}& 100\\ & \underset{―}{99}\\ & 1\end{array}}}$
The least three-digit numbers which are divisible by 11 is $100+(11-1)=110$.
Here, $110=11\times 10$.
$11\stackrel{90}{\overline{)\begin{array}{rl}& 999\\ & \underset{―}{99}\\ & 9\end{array}}}$
The highest three-digit numbers which are divisible by 11 is $999-9=990$.
Here, $990=11\times 90$.
Therefore, the multiplier for the three-digit numbers which are divisible by 11 runs from 10 to 90.So, there are in total $90-10+1=81$ three-digit numbers which are divisible by 11.The multiplier for the middle number of the divisible three-digit numbers will be ${\displaystyle \frac{10+90}{2}}={\displaystyle \frac{100}{2}}=50$.

Therefore, the middle number is $50\times 11=550$.

Note: The formula for middle point can also be applied for the numbers 110 and 990. We can also use the divisibility rule of 11 where if the difference of the sum of odd and even placed numbers is divisible by 11 then the main number is divisible by 11.