How Many Two Digit Numbers Are Divisible by 3 Using AP?

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

Here's how to determine that using the concept of an Arithmetic Progression (AP):

• Identifying the Sequence: The two-digit numbers divisible by 3 form an arithmetic progression. The smallest two-digit number divisible by 3 is 12, and the largest is 99. The sequence is: 12, 15, 18, ..., 99.

• Understanding AP: An arithmetic progression is a sequence where the difference between any two consecutive terms is constant. In this case, the common difference (d) is 3. The first term (a) is 12.

• Applying the Formula: The formula to find the nth term of an AP is:

  • a_n = a + (n - 1)d

  Where:

  • a_n is the nth term
  • a is the first term
  • n is the number of terms
  • d is the common difference

  We know that the last term (a_n) is 99. We can plug our values into this formula:

  • 99 = 12 + (n - 1)3

• Solving for 'n':

  1. Subtract 12 from both sides: 87 = (n - 1)3
  2. Divide both sides by 3: 29 = n - 1
  3. Add 1 to both sides: n = 30

• Conclusion: As stated in the provided reference, there are 30 two-digit numbers divisible by 3. This sequence forms an A.P., allowing us to utilize A.P. formulas.