Find the
(i) sum of those integers between 1 and 500 which are multiples of 2 as well as of 5.
(ii) sum of those integers from 1 to 500 which are multiples of 2 as well as of 5.
(iii) sum of those integers from 1 to 500 which are multiples of 2 or 5.

Explanation

(i) Since, multiples of 2 as well as of 5 = LCM of (2, 5)=10
` :. ` Multiples of 2 as well as of 5 between 1 and 500 is 10, 20, 30, …, 490
which form an AP with first term (a) = 10 and common difference (d) = 20 - 10 = 10
`n`th term `a_(n)` = Last term `(l)=490`
` :. ` Sum of `n` terms between 1 and 500,
`S_(n)=(n)/(2)[a+l] " "`...(i)
` :' a_(n)=a+(n-1)d=l`
`implies 10+(n-1)10=490`
`implies (n-1)10=480`
`implies n-1=48impliesn=49`
From Eq. (i), `S_(49)=(49)/(2)[10+490]`
` " "=(49)/(2)xx500`
` " "=49xx250=12250`
(ii) Same as part (i),
but multiples of 2 as well as of 5 from 1 to 500 is 10, 20, 30, ..., 500.
` :. a=10, d=10, a_(n)=l=500`
` :' a_(n)=a+(n-1)d=l`
`implies 500=10+(n-1)10`
`implies 490=(n-1)10`
`implies n-1=49impliesn=50`
` :' S_(n)=(n)/(2)(a+l)`
` implies S_(50)=(50)/(2)(10+500)= (50)/(2)xx510`
` " " =50xx255=12750`
(iii) Since, multiples of 2 or 5 = Multiple of 2 + Multiple of 5 - Multiple of LCM (2,5) I.e., 10.
` :. ` Multiples of 2 or 5 from 1 to 500
= List of multiple of 2 from 1 to 500 + List of multiple of 5 from 1 to 500
=(2, 4, 6, ..., 500)+(5, 10, 15, ..., 500)-(10, 20, ..., 500)
All of these list from an AP.
Now, number of terms in first list,
`500=2+(n_(1)-1)2implies 498=(n_(1)-1)2`
`implies n_(1)-1=249impliesn_(1)=250`
Number of terms in second list,
`500=5+(n_(2)-1)5implies 495=(n_(2)-1)5`
`implies99= (n_(2)-1)impliesn_(2)=100`
and number of terms in third list,
`500=10+(n_(3)-1)10implies 490=(n_(3)-1)10`
`implies n_(3)-1=49impliesn_(3)=50`
From Eq.(i), Sum of multiples of 2 or 5 from 1 to 500
= Sum of (2, 4, 6, ..., 500) + Sum of (5, 10, ..., 500) - Sum of (10, 20, ..., 500)
`=(n_(1))/(2)[2+500]+(n_(2))/(2)[5+500] - (n_(3))/(2)[10+500] " "[:' S_(n)=(n)/(2)(a+l)]`
`=(250)/(2)xx502+(100)/(2)xx505-(50)/(2)xx510`
`=250xx251+505xx50-25xx510=62750+25250-12750`
`=88000-12750=75250`