Step-by-Step Solution
Key Concept: The two‑digit numbers that are divisible by 3 form an arithmetic progression (AP) with common difference 3. By identifying the first and last terms of this AP, the number of terms can be obtained using the formula $n = \frac{l-a}{d}+1$.
1. Identify the range of two‑digit numbers: \(10 \leq N \leq 99\).
2. Find the smallest two‑digit number divisible by 3:
\[10 \equiv 1 \pmod{3},\; 11 \equiv 2 \pmod{3},\; 12 \equiv 0 \pmod{3}\]
Hence the first term \(a = 12\).
3. Find the largest two‑digit number divisible by 3:
\[99 \equiv 0 \pmod{3}\]
Hence the last term \(l = 99\).
4. Recognize the AP: The numbers \(12, 15, 18, \dots , 99\) constitute an AP with common difference \(d = 3\).
5. Use the AP term‑count formula:
\[n = \frac{l - a}{d} + 1 = \frac{99 - 12}{3} + 1 = \frac{87}{3} + 1 = 29 + 1 = 30\]
6. Conclusion: There are \(30\) two‑digit numbers that are divisible by 3.
Correct Answer: 30