Two-digit natural numbers are formed, with replacement, from the digits 0 through 9.How many two-digit odd numbers are possible?

Sure, here is the step by step solution:

Step 1: Identify the possible digits for each place in the two-digit number.

For a number to be odd, it must end in 1, 3, 5, 7, or 9. So, there are 5 possibilities for the units place.

Step 2: Identify the possible digits for the tens place.

Since zero cannot be the first digit of a two-digit number, the tens place can be any digit from 1 through 9. So, there are 9 possibilities for the tens place.

Step 3: Calculate the total number of possible two-digit odd numbers.

Since the digits for the tens and units places are chosen independently, the total number of possible two-digit odd numbers is the product of the number of possibilities for each place.

So, the total number of possible two-digit odd numbers is 9 (for the tens place) times 5 (for the units place), which equals 45.