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  • How many numbers lie between 10 and 300, which when divided by 4 leave a remainder 3?

How many numbers lie between 10 and 300, which when divided by 4 leave a remainder 3?

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\\$The numbers between 10 and 300 which when divided by 4 leave remainder 3 is-\\$ 11, 15, 19, 23, $ \ldots $ ., 299\\ a\textsubscript{1}=11, a\textsubscript{2}=15, a\textsubscript{3}=19\\ a\textsubscript{2} - a\textsubscript{1} = 15 - 11 = 4\\ a\textsubscript{3} - a\textsubscript{2} = 19 - 15 = 4\\ it is a\textsubscript{n} AP with a\textsubscript{1}=a = 11, d = 4\\ $ Let there are n terms in this AP\\$ a\textsubscript{n} = 299\\ a + (n - 1) \times d = 299 (using a\textsubscript{n} = a + (n-1)d)\\ 11 + (n - 1) \times 4 = 299\\ (n - 1) \times 4 = 299 - 11\\ (n - 1) \times 4 = 288\\ (n-1)=\frac{288}{4} \\ (n - 1) = 72\\ n = 72 + 1 \\ n = 73\\

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