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(D) The integers between $1$ and $500$ that are multiples of both $2$ and $5$ must be multiples of the $LCM(2, 5) = 10$.The sequence of these integers

Vedclass · Accepted Answer

$12250$

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(D) The integers between $1$ and $500$ that are multiples of both $2$ and $5$ must be multiples of the $LCM(2, 5) = 10$.The sequence of these integers is $10, 20, 30, \dots, 490$.This forms an Arithmetic Progression $(AP)$ where the first term $a = 10$, the common difference $d = 10$, and the last term $l = 490$.To find the number of terms $(n)$, we use the formula $a_n = a + (n - 1)d$:$490 = 10 + (n - 1)10$$480 = (n - 1)10$$n - 1 = 48$$n = 49$.The sum of an $AP$ is given by $S_n = \frac{n}{2}(a + l)$:$S_{49} = \frac{49}{2}(10 + 490)$$S_{49} = \frac{49}{2} 	imes 500$$S_{49} = 49 	imes 250 = 12250$.

Find the sum of those integers between $1$ and $500$ which are multiples of $2$ as well as of $5$.

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