Final answer:
To find the number of 2-digit numbers that are neither perfect squares nor divisible by 3, we can subtract the number of perfect squares and the number of numbers divisible by 3 from the total number of 2-digit numbers.
Step-by-step explanation:
To find the number of 2-digit numbers that are neither perfect squares nor divisible by 3, we can break the problem down into two parts. First, let's find the number of 2-digit perfect squares. The square roots of 10 and 100 are 3.16 and 10, respectively, so we can conclude that any perfect square between 10 and 100 will be a 2-digit number. There are 7 perfect squares in this range: 16, 25, 36, 49, 64, 81, and 100.
Next, let's find the number of 2-digit numbers divisible by 3. The largest 2-digit number divisible by 3 is 99. To find the number of terms in the sequence of 2-digit numbers divisible by 3, we can use the formula:
n = (last term - first term) / common difference + 1
In this case, the first term is 6 (the smallest 2-digit number divisible by 3) and the last term is 99. The common difference is 3 (since we're dealing with 2-digit numbers). Plugging these values into the formula, we get:
n = (99 - 6) / 3 + 1 = 32
Therefore, there are 32 2-digit numbers divisible by 3.
Now, to find the number of 2-digit numbers that are neither perfect squares nor divisible by 3, we can subtract the number of perfect squares (7) and the number of numbers divisible by 3 (32) from the total number of 2-digit numbers (90).
n = 90 - 7 - 32 = 51
So, there are 51 2-digit numbers that are neither perfect squares nor divisible by 3.