Question

The sum of integers from 1 to 300 that are divisible by 2 or 5 is :
1: 18150
2: 27150
3: 8850
4: 16750


Answer this question step by step choose correct option. ​

Answer 1

Solution :-

The integers from 1 to 300, which are divisible by 2, are 2, 4, 6… 300.

This forms an A.P. with both the first term and common difference equal to 2.

→ 300 = 2 + (n -1)2

→ n = 150

So,

→ S(150) = (n/2)[2a + (n - 1)d]

→ S(150) = (150/2)[2*2 + (150-1)2]

→ S(150) = 75[4 + 149*2]

→ S(150) = 75[ 4 + 298]

→ S(150) = 75 * 302

→ S(150) = 22,650

_____________

Now, The integers from 1 to 100, which are divisible by 5, are 5, 10… 300.

This forms an A.P. with both the first term and common difference equal to 5.

→ 300 = 5 + (n - 1)5

→ 5n = 300

→ n = 60

So ,

→ S(60) = (n/2)[2a + (n - 1)d]

→ S(60) = (60/2)[2*5 + (60-1)5]

→ S(60) = 30[10 + 59*5]

→ S(60) = 30[ 10 + 295]

→ S(60) = 30 * 305

→ S(60) = 9,150

_____________

And, in Last, The integers, which are divisible by both 2 and 5, are 10, 20, … 300.

This also forms an A.P. with both the first term 10 and common difference equal to 10.

→ 300 = 10 + (n - 1)10

→ 300 = 10n

→ n = 30

So,

→ S(30) = (n/2)[2a + (n - 1)d]

→ S(30) = (30/2)[2*10 + (30-1)10]

→ S(30) = 15[20 + 29*10]

→ S(30) = 15[ 20 + 290]

→ S(30) = 15 * 310

→ S(30) = 4,650

_____________

∴ Required sum = 22650 + 9150 - 4650 = 27,150 (Option 2) (Ans.)

Hence, the sum of the integers from 1 to 300, which are divisible by 2 or 5, is 27,150.