The sum of integers from 1 to 50 that are div | vedclass

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(B) Integers divisible by both $2$ and $3$ are divisible by their least common multiple,which is $6$. The integers from $1$ to $50$ divisible by $6$ a

Vedclass · Accepted Answer

$6^3$

Vedclass · Answer

(B) Integers divisible by both $2$ and $3$ are divisible by their least common multiple,which is $6$. The integers from $1$ to $50$ divisible by $6$ are $6, 12, 18, \dots, 48$. This forms an Arithmetic Progression $(AP)$ where the first term $a = 6$,the last term $a_n = 48$,and the common difference $d = 6$. Using the formula $a_n = a + (n - 1)d$: $48 = 6 + (n - 1)6$ $42 = (n - 1)6$ $n - 1 = 7 \Rightarrow n = 8$. The sum $S_n$ of an $AP$ is given by $S_n = \frac{n}{2}(a + a_n)$: $S_8 = \frac{8}{2}(6 + 48) = 4(54) = 216$. Since $216 = 6^3$,the correct option is $B$.

The sum of integers from $1$ to $50$ that are divisible by both $2$ and $3$ is

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