The use of arithmetic formula will help you find the sum of such integers quickly.

We want all three digit integers that gives remainder of 2 upon dividing by 3.

some important points regarding this problem.

1) there are only 3 possible remainders for any integer upon dividing by 3 which are 0, 1, 2. You will get the remainder of 2 when the integer being divided is 1 less than the integer that is evenly divisible by 3. First of such kind of integer in 101 and last is 998 (since 102 and 999 are evenly divisible by 3)

2) the next integer that gives remainder of 2 when divided by 3 after 101 will be 101+3 i.e. 104. We can form here the sequence of all the integers that gives remainder of 2 when divided by 3 in which every integer will be 3 greater than the previous integer. 101, 104, 107, 110, ...........,998.

3) So we have arithmetic progression with first term 101, last term 998, and common difference 3.

4) Formula to calculate number of terms in the sequence with common difference 3 is
(i) Including both ends [(last term - first term)/3] + 1
(ii) Including only one end [(last term - first term)/3]
(iii) Excluding both ends [(last term - first term)/3] - 1

5) Since we want both ends to be included in the counting, we will use the first case. So number of terms is [(998 - 101)/3] +1 ---------> 299 + 1 -------> 300

6) Sum of arithmetic progression with first term a, last term l and number of terms n will be n(a+l)/2 -------> 300(1099)/2 ---------> 164850

Hope that's clear.
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What is this AP formula? What does AP stand for? Thanks.

Formula for n th term = a+(n-1)d
So here a = 101, d=3,n th term = 998

998 = 101 + (n-1)3 ..on solving n = 300 ....so there r 300 termsin the series...

My Thought : Integer Rules
My Understanding of this Question
Find the number , Upon Sum of 3 Digits of a number Gives a Reminder 2 when it is Divided by 3
Seeing the Options After Dividing an Finding the Reminder of 2
My Answer was C
My Answer was Wrong

But I Don't Understand Why there is need of AP , What's the Question is Testing

Pls Help:)
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Let me ask you a question: how many leftover apples would you have if you had 2 apples and wanted to distribute in 7 baskets evenly? Each basket would get 0 apples and 2 apples would be leftover (remainder).

When a divisor is more than dividend, then the remainder equals to the dividend, for example:
3 divided by 4 yields the reminder of 3: \(3=4*0+3\);
9 divided by 14 yields the reminder of 9: \(9=14*0+9\);
1 divided by 9 yields the reminder of 1: \(1=9*0+1\).


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Reading through question is key here :
Given sum of three digit numbers : the smallest three digit is 100 but the condition is when number is divided by 3 the remainder must be 2 hence the series will begin from 101 till the largest three digit number (when divided by 3 gives remainder 2) i.e. 998.

to find the sum of AP : N/2 (a + l) where N is number of terms , a is first term here its 101 and l is last term of series here its 998.
to calculate N = (last term - first number)/3 + 1
hence N = 300
Sum = 300/2 (101 + 998)
Hence answer is B

Queries are encouraged

In step 3, why are we dividing by 3 + 1 ?

The following post might help: how-many-multiples-of-4-are-there-between-12-and-94862.html#p730075
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Another approach to this question =>
clearly the question asks us to find the sum of series => 101+104+107....998
here the number of terms => [(first term - last term)/3]+ 1 = 300
Mean of any AP series is the average of the first term and the last term
hence here the mean => (101+998)/2 = 1099/2
now mean = sum of all the terms / total no. of terms
hence the sum of all the terms = mean * total number of terms.
hence the sum = (1099/2) *300 = 164850
hence B
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As we can see, the integers in question are: 101, 104, 107, …, 998. Since we are calculating a sum of an evenly-spaced set of integers, we can use the following equation to determine the sum:

sum = average x quantity

The smallest 3-digit integer that leaves a remainder of 2 when divided by 3 is 101, and the largest is 998. We can use these values to determine the average and quantity. Let’s start by determining the average:

avg = (largest integer + smallest integer)/2

avg = ( 998 + 101)/2

avg = 1,099/2

Next we can determine the quantity:

quantity = (largest integer - smallest integer)/3 + 1

quantity = (998 - 101)/3 + 1 = 299 + 1 = 300

Now we can determine the sum:

sum = (1,099/2)(300) = 164,850

Answer: B
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