Question

Find the total number of three-digit numbers, the sum whose digit is even.

Answer 1

In order to find the number of three-digit numbers, the sum whose digit is even, we must check out for the rules of sum of even numbers, sum of odd numbers and sum of even and odd numbers. From these rules, we must be considering the favourable rule and we must check all the three-digit numbers for our required answer.

Complete step by step answer:
Now let us have a brief regarding even and odd numbers. Even numbers are those that are divisible by $2$ which simply means that it can be divided into two equal groups. Even numbers are those that end with $0,2,4,6,8$. Odd numbers are those that end with the following digits: $1,3,5,7,9$. The only even number that is a prime number is $2$. We can identify an odd number by either checking for the digit in one place or by trying to divide it into two equal groups.
Now let us find the total number of three-digit numbers, the sum whose digit is even.
So now we will be considering the numbers where all the digits are even.
We know that $even+even+even=even$
In this case, we can select the digit in hundreds of places in four ways. They are $2,4,6,8$.
The digits in the tens and the ones place can be selected in five ways. They include $0,2,4,6,8$.
Now let us calculate how much such three digit numbers would form.
$\Rightarrow 4\times 5\times 5=100$
$\therefore$ $100$ such numbers exist.
Now let us consider another case where one of the digits would be even.
We consider this case because $odd+odd+even=even$.
We can select the digit in hundreds of places in five ways. If at all, the digit in the hundred’s place would be even, it can selected in four ways.
The total numbers such exist would be $4\times 5\times 5=100$
For instance, an odd number occurs in the hundred’s place, then the number of such numbers would be $5\times 5\times 5=125$.
In this case, we cannot be sure regarding the positions of the even and the odd numbers. The positions can interchange.
So considering those numbers also, we get the total count as $125\times 2=250$.
$\therefore$ The numbers in the second case considered would be $250+100=350$.
Now, let us calculate both the count of numbers from case one and case two.
We get,
$350+100=450$.
$\therefore$ The total number of three-digit numbers, the sum whose digit is even is $450$.

Note: We must always have a note of the sum rule of the odd numbers and even numbers as per the requirement. We can apply the concept of even and odd numbers in digital circuit systems, binary codes, in geometry and many more.