Find the sum of integers from 1 to 100 that a | vedclass

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(A) Let $S_2$ be the sum of integers divisible by $2$ and $S_5$ be the sum of integers divisible by $5.$The integers divisible by $2$ are $2, 4, 6, \l

Vedclass · Accepted Answer

$3050$

Vedclass · Answer

(A) Let $S_2$ be the sum of integers divisible by $2$ and $S_5$ be the sum of integers divisible by $5.$The integers divisible by $2$ are $2, 4, 6, \ldots, 100.$ This is an $A.P.$ with $a=2, d=2, n=50.$$S_2 = \frac{50}{2}(2 + 100) = 25 	imes 102 = 2550.$The integers divisible by $5$ are $5, 10, 15, \ldots, 100.$ This is an $A.P.$ with $a=5, d=5, n=20.$$S_5 = \frac{20}{2}(5 + 100) = 10 	imes 105 = 1050.$The integers divisible by both $2$ and $5$ (i.e.,divisible by $10$) are $10, 20, \ldots, 100.$ This is an $A.P.$ with $a=10, d=10, n=10.$$S_{10} = \frac{10}{2}(10 + 100) = 5 	imes 110 = 550.$By the Principle of Inclusion-Exclusion,the required sum is $S_2 + S_5 - S_{10}.$Required sum $= 2550 + 1050 - 550 = 3050.$

Find the sum of integers from $1$ to $100$ that are divisible by $2$ or $5.$

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