Find the sum of integers between 100 and 200 that are
(i) divisible by 9, (ii) not divisible by 9.
(i) 1683 (ii) 13167
Step (i): Integers divisible by 9
1. First, write down the numbers between 100 and 200 which are divisible by 9.
The smallest number greater than 100 divisible by 9 is 108, and the largest below 200 is 198.
So the numbers are: 108, 117, 126, …, 198.
2. This is an arithmetic progression (AP) with first term \(a = 108\), last term \(l = 198\), and common difference \(d = 9\).
3. To find how many terms there are, use the AP formula: \(n = \dfrac{l - a}{d} + 1\).
\(n = \dfrac{198 - 108}{9} + 1 = \dfrac{90}{9} + 1 = 10 + 1 = 11\).
4. The sum of an AP is given by: \(S = \dfrac{n}{2}(a + l)\).
\(S = \dfrac{11}{2}(108 + 198) = \dfrac{11}{2}(306) = 11 \times 153 = 1683\).
So, the sum of numbers divisible by 9 is 1683.
Step (ii): Integers not divisible by 9
1. Integers strictly between 100 and 200 are: 101, 102, …, 199.
This is another AP with first term \(a = 101\), last term \(l = 199\), and number of terms \(n = 99\).
2. The sum of these 99 numbers is:
\(S = \dfrac{n}{2}(a + l) = \dfrac{99}{2}(101 + 199) = \dfrac{99}{2}(300) = 99 \times 150 = 14850\).
3. Out of this total sum, the part divisible by 9 (already found in step (i)) is 1683.
4. Therefore, the sum of integers not divisible by 9 is:
\(14850 - 1683 = 13167\).
So, the sum of numbers not divisible by 9 is 13167.