Question

How many two digit numbers are divisible by 7?

• A. 10
• B. 11
• C. 12
• D. 13

Answer 1

The Correct Option is D

To solve the problem, we need to find how many two-digit numbers are divisible by 7.

1. Define the range of two-digit numbers:
Two-digit numbers range from 10 to 99.

2. Find the smallest two-digit number divisible by 7:
The smallest two-digit number is 10.
Divide 10 by 7: \( \frac{10}{7} \approx 1.42 \)
Next multiple of 7 is: \( 7 \times 2 = 14 \)

3. Find the largest two-digit number divisible by 7:
The largest two-digit number is 99.
Divide 99 by 7: \( \frac{99}{7} \approx 14.14 \)
Largest multiple of 7 within range: \( 7 \times 14 = 98 \)

4. Count the number of multiples from 14 to 98:
This is an arithmetic sequence: \( 14, 21, 28, ..., 98 \)
Use the formula for the nth term of an AP:
\( a_n = a + (n - 1)d \), where \( a = 14 \), \( d = 7 \)
We solve: \( 14 + (n - 1) \cdot 7 = 98 \)
\( (n - 1) \cdot 7 = 84 \Rightarrow n - 1 = 12 \Rightarrow n = 13 \)

Final Answer:
The correct answer is option (D): 13.