Finding the Count of Two-Digit Numbers Divisible by 3

The question asks us to find the total number of two-digit numbers that can be divided by 3 without leaving a remainder. These are the two-digit multiples of 3.

Identifying the Range of Two-Digit Numbers

Two-digit numbers are integers starting from 10 and going up to 99. Numbers like 1, 5, 9 are single-digit, and numbers like 100, 105 are three-digit or more.

So, the range of numbers we are considering is 10, 11, 12, ..., 98, 99.

Finding the Smallest and Largest Two-Digit Multiples of 3

• Let's look for the first number in this range (10 to 99) that is divisible by 3.
• 10 is not divisible by 3.
• 11 is not divisible by 3.
• 12 is divisible by 3 ($12 \div 3 = 4$). So, the smallest two-digit number divisible by 3 is 12.

• Now, let's look for the last number in this range (10 to 99) that is divisible by 3.
• 99 is divisible by 3 ($99 \div 3 = 33$). So, the largest two-digit number divisible by 3 is 99.

Listing the Two-Digit Numbers Divisible by 3

The two-digit numbers divisible by 3 form a sequence: 12, 15, 18, 21, ..., 96, 99.

This sequence is an arithmetic progression (AP) because the difference between consecutive terms is constant (which is 3, as they are multiples of 3).

• First term ($a$) = 12
• Common difference ($d$) = 3
• Last term ($a_n$) = 99

Calculating the Number of Terms (Count)

We can use the formula for the nth term of an arithmetic progression: $a_n = a + (n-1)d$, where $n$ is the number of terms.

Substitute the values we found:

\( 99 = 12 + (n-1)3 \)

Subtract 12 from both sides:

\( 99 - 12 = (n-1)3 \)

\( 87 = (n-1)3 \)

Divide by 3:

\( \frac{87}{3} = n-1 \)

\( 29 = n-1 \)

Add 1 to both sides to find \(n\):

\( n = 29 + 1 \)

\( n = 30 \)

So, there are 30 two-digit numbers divisible by 3.

Alternative Method: Using Division

We can also find the count by considering the multiples of 3 up to 99 and subtracting the multiples of 3 that are single-digit (or zero).

The number of multiples of 3 up to 99 is $\lfloor 99/3 \rfloor = 33$. These are 3, 6, 9, ..., 99.

The single-digit multiples of 3 are 3, 6, 9. The number of single-digit multiples of 3 is $\lfloor 9/3 \rfloor = 3$.

The number of two-digit multiples of 3 is the total number of multiples up to 99 minus the number of multiples up to 9.

Count = (Number of multiples of 3 up to 99) - (Number of multiples of 3 up to 9)

Count = $33 - 3 = 30$.

Both methods show that there are 30 two-digit numbers divisible by 3.