Question

Find the sum of all three digit numbers which are divisible by 3 but not divisible by 5

Answer 1

Answer: The sum of all three digit numbers which are divisible by 3 but not divisible by 5 is 132300.

Step-by-step explanation:

Since we have given that

All three digit numbers which are divisible by 3 but not divisible by 5.

So, First we will write all the three digit numbers which are divisible by 3.

$102,105,.....999$

So, we will find the number of terms from 102 to 999.

$a_n=a+(n-1)d\\\\999=102+(n-1)3\\\\999-102=3(n-1)\\\\897=3(n-1)\\\\\frac{897}{3}=n-1\\\\299=n-1\\\\n=300$

And sum of all the above numbers is given by

$S_{300}=\frac{n}{2}(a+a_n)=\frac{300}{2}(102+999)\\\\S_{300}=150\times (1101)\\\\S_{300}=165150$

Now, we will find the series which are divisible by 3 and divisible by 5 i.e. divisible by 15.

105,120,........990

So, number of terms will be

$a_n=a+(n-1)d\\\\990=105+(n-1)\times 15\\\\990-105=(n-1)\times 15\\\\885=(n-1)\times 15\\\\\frac{885}{15}=n-1\\\\59=n-1\\\\n=59+1=60$

So, the sum will be

$S_{60}=\frac{n}{2}(a+a_n)\\\\S_{60}=\frac{60}{2}(105+990)\\\\S_{60}=30(1095)\\\\S_{60}=32850$

So, the sum of all three digit numbers which are divisible by 3 but not divisible by 5 is given by

$165150-32850\\\\=132300$