How many two-digit numbers are divisible by 3?
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Step-by-Step Solution
Step 1: Identify the first two-digit number divisible by 3
The smallest two-digit number is 10. We check if 10 is divisible by 3. Since 10=3×3+1, it is not. The next multiple of 3 after 9 (which is a one-digit number) is 3×4=12. So, 12 is the first two-digit number divisible by 3.
Step 2: Identify the last two-digit number divisible by 3
The largest two-digit number is 99. We check if 99 is divisible by 3. Since 99=3×33, it is perfectly divisible by 3. Thus, 99 is the last two-digit number divisible by 3.
Step 3: Form an Arithmetic Progression (AP) and identify its components
The sequence of two-digit numbers divisible by 3 forms an Arithmetic Progression (AP) where the first term (a) is 12, the common difference (d) is 3, and the last term (an) is 99. We need to find the number of terms (n) in this AP.
Step 4: Apply the nth term formula for an AP
The formula for the nth term of an Arithmetic Progression is an=a+(n−1)d. We will substitute the values we found into this formula to solve for n.
Step 5: Substitute values and solve for n
Substituting a=12, d=3, and an=99 into the formula an=a+(n−1)d, we get 99=12+(n−1)3. Subtracting 12 from both sides gives 87=(n−1)3. Dividing by 3 yields 29=n−1. Finally, adding 1 to both sides gives n=30. Therefore, there are 30 such numbers.