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Question
Find the sum of all those integers between 100 and 800 each of which on division by 16 leaves the remainder 7.
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Solution
The sum of all those integers between 100 and 800 each of which on division by 16 leaves the remainder 7 are:
103, 119...791
Here, we have:
a = 103
d = 16
\[a_n = 791\]
\[\text { We know }: \]
\[ a_n = a + (n - 1)d\]
\[ \Rightarrow 791 = 103 + (n - 1) \times 16\]
\[ \Rightarrow 688 = 16n - 16\]
\[ \Rightarrow 704 = 16n\]
\[ \Rightarrow 44 = n\]
\[\text { Also }, S_n = \frac{n}{2}[2a + (n - 1)d]\]
\[ \Rightarrow S_{44} = \frac{44}{2}[2 \times 103 + (44 - 1) \times 16]\]
\[ \Rightarrow S_{44} = 22 [206 + 688]\]
\[ \Rightarrow S_{44} = 22 \times 894 = 19668\]
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