How Many Numbers of Two Digit are Divisible by 3?

Answer: There are 30 two-digit numbers that are divisible by 3.

Step-by-Step Solution

We are asked to find how many two-digit numbers are divisible by 3.

Step 1: Identify the range of two-digit numbers

Two-digit numbers start from 10 and go up to 99.

So, we are looking for all numbers between 10 and 99 that are divisible by 3.

Step 2: Use the concept of Arithmetic Progression (A.P.)

The numbers divisible by 3 in this range form an arithmetic progression:

• The first two-digit number divisible by 3 is 12
  (Since 3 × 4 = 12 and 12 is the first two-digit multiple of 3)
• The last two-digit number divisible by 3 is 99
  (Because 3 × 33 = 99, which is the last two-digit multiple of 3)

Now, we have an A.P.: 12, 15, 18, …, 99 

Here,

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

Step 3: Use the A.P. formula to find the number of terms

The formula to find the number of terms (n) in an A.P. is:

Substitute the values:

Final Answer:

There are 30 two-digit numbers divisible by 3.

Quick Tip for Class 10 Students

Instead of checking each number individually, you can apply the A.P. formula anytime you need to count numbers divisible by a fixed number within a range. It’s a major time-saver, especially in competitive exams or board tests!

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