Counting Two-Digit Numbers Divisible by 3

To determine how many two-digit numbers are divisible by 3, follow these steps:

Step 1: Identify the Smallest and Largest Two-Digit Numbers Divisible by 3

Finding the Smallest Two-Digit Number Divisible by 3:

1. The smallest two-digit number is 10.
2. To find the smallest two-digit number divisible by 3, divide 10 by 3: $$\frac{10}{3} \approx 3.33$$ Round up to the next whole number, which is 4.
3. Multiply 4 by 3 to get: $$4 \times 3 = 12$$ Therefore, 12 is the smallest two-digit number divisible by 3.

Finding the Largest Two-Digit Number Divisible by 3:

1. The largest two-digit number is 99.
2. To confirm that 99 is divisible by 3, divide 99 by 3: $$\frac{99}{3} = 33$$ Since 33 is an integer, 99 is divisible by 3.

Step 2: Formulate the Arithmetic Sequence

The two-digit numbers divisible by 3 form an arithmetic sequence with:

• The first term $a = 12$
• The common difference $d = 3$

Step 3: Calculate the Number of Terms in the Sequence

To find the number of terms ($n$) in the sequence, use the formula for the $n$-th term of an arithmetic sequence:

$$a_n = a + (n-1)d$$

Where:

• $a_n$ is the last term (99 in this case)
• $a$ is the first term (12)
• $d$ is the common difference (3)

Substitute the values into the formula and solve for $n$:

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

Subtract 12 from both sides:

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

Divide both sides by 3:

$$29 = n-1$$

Solve for $n$:

$$n = 29 + 1$$ $$n = 30$$

Conclusion

There are 30 two-digit numbers that are divisible by 3. This result is obtained by analyzing the arithmetic sequence formed by these numbers, showing the practical application of arithmetic series in solving numerical problems.