How Many Two-Digit Numbers are Divisible by 2

Introduction

The numbers with only two place values i.e the tens place and the units place are said to be two-digit numbers. Two-digit numbers range from 10 to 99 i.e the two-digit numbers start from 10 and end at 99. Before 10, we have one-digit numbers and after 99 we have three-digit numbers. Also, we can say the largest two-digit number is 99 and the smallest two-digit number is 10. Between 10 and 99, there are numbers which are either even or odd, so the numbers which are even are divisible by 2, for example, 10,12,14 etc are all even numbers hence these are said to be numbers divisible by 2.

Even Numbers

Those numbers are said to be even numbers which are exactly divisible by 2 i.e the numbers which on dividing by 2 give 0 as the remainder. In other words, we can say Even numbers are numbers that can be divided into two equal groups or pairs.

For example-

Let's say we have a two-digit number 54

\begin{equation}

54/2=27

\end{equation}

So we get 27 as the quotient and 0 as the remainder. Hence 54 is an even number and is, therefore, divisible by 2.

Let's take some more examples:

Example 1: 26

\begin{equation}

26 / 2=13

\end{equation}

Here on dividing 26 by 2, we get 13 as the Quotient and 0 as the Remainder so 26 is an even number and is, therefore, divisible by 2.

Example 2: 17

\begin{equation}

17 / 2

\end{equation}

\begin{equation}

Q=8, R=1

\end{equation}

Here On dividing 17 by 2, we get 8 as the Quotient and 1 as the Remainder so 17 is not an even number and is, therefore, not divisible by 2. These numbers which are not divisible by 2 are said to be Odd Numbers.

Let's dive into learning about odd numbers.

Odd Numbers

Those numbers are said to be odd numbers which by dividing by 2 do not give 0 as the remainder. In other words, we can say the numbers which cannot be divided into equal pairs are called Odd Numbers. For example, 53,23,35,47,69 etc are all odd numbers.

There can be various other definitions of odd numbers:

The numbers which have their unit's place as either 1,3,5,7,9 are said to be Odd Numbers. For example, 3,45,57,69 have 3,5,7,9 as their units place respectively so these are all considered odd numbers.

The numbers which are not even numbers are said to be Odd Numbers.

Let's see some examples of Odd Numbers:

Example 1: 25

There are various ways to check whether a number is odd or even.

Method 1:

Since the unit's place of 25 is 5 which is an odd number, hence we can say 25 is an Odd Number.

Method 2:

Another way is to divide the number by 2, if the Remainder is not equal to 0 then the number is said to be an Odd Number.

\begin{equation}

25 / 2

\end{equation}

\begin{equation}

Q=12, R=1

\end{equation}

Here On dividing 17 by 2, we get 8 as the Quotient and 1 as the Remainder.

So since we are getting Remainder which is not equal to 0 i.e 1 so 25 is an Odd Number.

Example 2: 57

Since the last digit or the unit's digit or the one's place of the given number 57 is 7 which is an odd number, So the number 57 is also an Odd Number.

\begin{equation}

57 / 2

\end{equation}

\begin{equation}

Q=28, R=1

\end{equation}

Also on dividing 57 by 2 we get 28 as Quotient and 1 as Remainder so 57 is an odd number since the remainder is not equal to 0.

Count of two-digit numbers divisible by 2

As we have studied, the numbers which are even are divisible by 2.

The smallest even two-digit number is 10 and the largest two-digit even number is 98.

So to understand a total number of two-digit numbers which are exactly divisible by 2 we will have to use a formula.

The formula for a total count of two-digit numbers is:

\begin{equation}

l=a+(n-1) d

\end{equation}

Let’s understand the meaning of the given terms in the above equation:

Here the variable ‘l’ represents the last term.

Here the variable ‘a’ represents the first term.

Here the variable ‘n’ represents the number of the terms in the sequence.

Here the variable ‘d’ represents the difference between the two consecutive terms.

Now we have to deal with ‘n’ as it provides the total number of two digits numbers.

We know the smallest or the first two-digit number which is divisible by 2 is 10 and the largest two-digit number divisible by 2 is 98.

Also, the difference between two consecutive two-digit numbers divisible by 2 is two i.e

10,12,14 etc are the numbers divisible by 2 and we can see the difference between two consecutive numbers is 2.

So our sequence of two-digit even numbers is like this:

10, 12, 14, ………………………………………………………, 98

So now assigning all values to the formula:

\begin{equation}

\begin{aligned}

& l=a+(n-1) d \\

& 98=10+(n-1) 2 \\

& 98=10+2 n-2 \\

& 98=8+2 n \\

& 98-8=2 n \\

& 90=2 n \\

& 90 / 2=n \\

& 45=n

\end{aligned}

\end{equation}

Hence by using the formula we have found that the total number of two-digit numbers which are divisible by 2 is 45.

Therefore there are 45 two-digit numbers which are divisible by 2.

Let's have a view of those 45 two-digit numbers which are divisible by 2:

Conclusion

Following points we had learnt:

• Two-digit numbers are the numbers which have two place values i.e the tens place and the one's place or the units place.

• Even Numbers are the numbers which are divisible by 2 or we can say the numbers which have their unit's place as either 0, or 2,4,6,8.

• Odd Numbers are the numbers which are not divisible by 2 or we can say the numbers which have their unit's place as either 1,3,5,7,9.

• There are a total of 45 two-digit numbers which are divisible by 2.

• The formula for calculating the total number of two-digit numbers is

\begin{equation}

l=a+(n-1) d

\end{equation}