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(B) Let $A$ be the set of integers divisible by $2$ and $B$ be the set of integers divisible by $5$ in the range $[1, 100]$.We need to find the sum of

Vedclass · Accepted Answer

$3050$

Vedclass · Answer

(B) Let $A$ be the set of integers divisible by $2$ and $B$ be the set of integers divisible by $5$ in the range $[1, 100]$.We need to find the sum of elements in $A \cup B$,which is given by $S(A \cup B) = S(A) + S(B) - S(A \cap B)$.$S(A)$ is the sum of multiples of $2$: $2, 4, 6, \dots, 100$. This is an $A$.$P$. with $n=50$,$a=2$,$l=100$. $S(A) = \frac{50}{2}(2 + 100) = 25 	imes 102 = 2550$.$S(B)$ is the sum of multiples of $5$: $5, 10, 15, \dots, 100$. This is an $A$.$P$. with $n=20$,$a=5$,$l=100$. $S(B) = \frac{20}{2}(5 + 100) = 10 	imes 105 = 1050$.$S(A \cap B)$ is the sum of multiples of $10$ ($LCM$ of $2$ and $5$): $10, 20, 30, \dots, 100$. This is an $A$.$P$. with $n=10$,$a=10$,$l=100$. $S(A \cap B) = \frac{10}{2}(10 + 100) = 5 	imes 110 = 550$.Therefore,$S(A \cup B) = 2550 + 1050 - 550 = 3050$.

The sum of all integers from $1$ to $100$ that are divisible by $2$ or $5$ is...........

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