To find the sum of numbers from 1 to 100 that are not divisible by 3 and 5, we use the principle of inclusion-exclusion.
Step 1: Calculate the sum of all numbers from 1 to 100.
The sum of the first n natural numbers is given by S=2n(n+1).
Stotal=2100(100+1)=50×101=5050
Step 2: Find the sum of numbers divisible by 3.
The numbers are 3,6,9,…,99. This is an A.P. where a=3,l=99,n=33.
S3=233(3+99)=233(102)=33×51=1683
Step 3: Find the sum of numbers divisible by 5.
The numbers are 5,10,15,…,100. This is an A.P. where a=5,l=100,n=20.
S5=220(5+100)=10×105=1050
Step 4: Find the sum of numbers divisible by both 3 and 5 (i.e., divisible by 15).
The numbers are 15,30,45,60,75,90. Here a=15,l=90,n=6.
S15=26(15+90)=3×105=315
Step 5: Calculate the sum of numbers divisible by 3 or 5.
Using the formula S(3∪5)=S3+S5−S15:
Sdivisible=1683+1050−315=2418
Step 6: Final Calculation.
The sum of numbers not divisible by 3 and 5 is:
Required Sum=Stotal−Sdivisible
Required Sum=5050−2418=2632
Correct Option:
(c) 2632
