How many two-digit numbers are divisible by 3?

To determine how many two-digit numbers are divisible by 3, we start by identifying the smallest and largest two-digit numbers that meet this criterion. The smallest two-digit number divisible by 3 is 12, and the largest two-digit number divisible by 3 is 99.

Analyzing the Problem

These numbers form an arithmetic progression (A.P.), where:

• The first term (a) is 12.
• The common difference (d) is 3 because consecutive numbers divisible by 3 differ by 3.
• The nth term (l) is 99.

To find how many two-digit numbers are divisible by 3, we use the formula for the nth term of an A.P.:

Formula: l = a + (n - 1) × d

Substituting the values:

99 = 12 + (n - 1) × 3

Simplify the equation:

99 - 12 = (n - 1) × 3
87 = (n - 1) × 3
n - 1 = 87 ÷ 3
n - 1 = 29
n = 30

Conclusion

Therefore, there are 30 two-digit numbers that are divisible by 3. These numbers include 12, 15, 18, ..., up to 99. The solution confirms that the sequence contains 30 terms, so the correct answer is option (3).

Additional Explanation

To summarize, the question "how many two-digit numbers are divisible by 3" is solved by identifying the A.P., applying the nth term formula, and calculating the total terms. The key steps ensure accuracy and show that the answer to "how many two-digit numbers are divisible by 3" is indeed 30.