Two-digit natural numbers are formed, with replacement, from the digits 0 through 9.How many numbers are greater than 65?

1. Break Down the Problem

To find the two-digit natural numbers greater than 65, we need to consider the possible combinations of the two digits while ensuring the first digit is in the range of 6-9 (since we are looking for numbers greater than 65).

2. Relevant Concepts

A two-digit number can be represented as $\text{AB}$, where:

• $\text{A}$ is the first digit (tens place) and can take values from 6 to 9.
• $\text{B}$ is the second digit (units place) and can take values from 0 to 9.

3. Analysis and Detail

We analyze the choices for $\text{A}$ and $\text{B}$:

• If $\text{A} = 6$: Possible $\text{B}$ values for numbers greater than 65 are 6, 7, 8, 9. There are 4 options: 66, 67, 68, 69.
• If $\text{A} = 7$: $\text{B}$ can be anything from 0 to 9. This gives us 10 options: 70, 71, 72, 73, 74, 75, 76, 77, 78, 79.
• If $\text{A} = 8$: Similarly, $\text{B}$ can be 0 to 9, giving another 10 options: 80, 81, 82, 83, 84, 85, 86, 87, 88, 89.
• If $\text{A} = 9$: Again, $\text{B}$ ranges from 0 to 9, resulting in 10 options: 90, 91, 92, 93, 94, 95, 96, 97, 98, 99.

4. Verify and Summarize

Now, we summarize the total possibilities:

• For $\text{A} = 6$: 4 options
• For $\text{A} = 7$: 10 options
• For $\text{A} = 8$: 10 options
• For $\text{A} = 9$: 10 options

Calculating the total: $\text{Total} = 4 + 10 + 10 + 10 = 34$

Final Answer

The total number of two-digit natural numbers greater than 65 is $\boxed{34}$.