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Question: The sum of the integers from 1 to 100 which are not divisible by 3 or 5 is....

The sum of the integers from 1 to 100 which are not divisible by 3 or 5 is.

A

2489

B

4735

C

2317

D

2632

Answer

2632

Explanation

Solution

Let S=1+2+3+…………+100S = 1 + 2 + 3 + \ldots \ldots \ldots \ldots + 100

=1002(1+100)=50(101)=5050= \frac { 100 } { 2 } ( 1 + 100 ) = 50 ( 101 ) = 5050

Let S1=3+6+9+12+………..+99S _ { 1 } = 3 + 6 + 9 + 12 + \ldots \ldots \ldots . . + 99

=

= 3.332(1+33)=99×17=16833 . \frac { 33 } { 2 } ( 1 + 33 ) = 99 \times 17 = 1683

Let

=

=

Let S3=15+30+45+………+90S _ { 3 } = 15 + 30 + 45 + \ldots \ldots \ldots + 90

= 15(1+2+3+……..+6)15 ( 1 + 2 + 3 + \ldots \ldots . . + 6 )

= 15⋅62(1+6)=45×7=31515 \cdot \frac { 6 } { 2 } ( 1 + 6 ) = 45 \times 7 = 315

∴ Required sum =S−S1−S2+S3S - S _ { 1 } - S _ { 2 } + S _ { 3 }

=5050−1683−1050+3155050 - 1683 - 1050 + 315= 2632.