How many integer palindromes are between 10,000 and 70,000 and how many of those are even?

the numbers are

1xxx1
2xxx2
3xxx3
4xxx4
5xxx5
6xxx6
Since each of the x's can be any digit, but the first and last must be the same, there are 100 palindromes in each group.

six groups means 600 palindromes.

3 of the groups are even.
So, that makes 300 even palindromes.

To find the number of integer palindromes between 10,000 and 70,000, we need to count how many numbers have the same digits when read forward and backward.

Step 1: Determine the range of possible digits for each place value:
Since we are looking for integers between 10,000 and 70,000, the first digit (tens of thousands place) can be any digit between 1 and 7, inclusive (i.e., 1, 2, 3, 4, 5, 6, or 7).
The second digit (thousands place) can be any digit between 0 and 9, inclusive (i.e., 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9).
The third digit (hundreds place) can be any digit between 0 and 9, inclusive.
The fourth digit (tens place) can be any digit between 0 and 9, inclusive.
The fifth digit (ones place), just like the first digit, can be any digit between 1 and 7, inclusive.

Step 2: Count the number of palindromes:
To form a palindromic number, the first and last digits must be the same, and the second and fourth digits must be the same. Since there are 7 choices for the first digit and 10 choices for the second, third, and fourth digits, the total number of palindromes with these criteria is 7 * 10 * 10 * 10 * 7 = 49,000.

Step 3: Count the number of even palindromes:
Since we are looking for even palindromes, the last digit (ones place) must be even. There are 5 even digits to choose from (i.e., 0, 2, 4, 6, or 8) out of the 7 available choices for both the first and last digits. So the total number of even palindromes is 5 * 10 * 10 * 10 * 5 = 25,000.

Therefore, there are 49,000 integer palindromes between 10,000 and 70,000, and out of those, 25,000 are even.

To find the number of integer palindromes between 10,000 and 70,000, we need to identify the conditions that a number must satisfy to be a palindrome.

A palindrome is a number that reads the same forwards and backward. In this case, it means that the thousands digit must be the same as the ones digit, and the hundreds digit must be the same as the tens digit.

Let's break it down step by step to determine the number of palindromes between 10,000 and 70,000:

1. Consider the thousands digit:
The thousands digit can range from 1 to 7, as the number must be between 10,000 and 70,000.

2. Consider the hundreds digit:
The hundreds digit can range from 0 to 9 since it can be any number.

3. Consider the tens digit:
The tens digit must be the same as the thousands digit, so it is determined by the thousands digit.

4. Consider the ones digit:
The ones digit must be the same as the hundreds digit, so it is determined by the hundreds digit.

Now, let's calculate the number of palindromes:

1. The thousands digit can be any of the numbers {1, 2, 3, 4, 5, 6, 7}, so there are 7 choices.

2. The hundreds digit can be any of the numbers {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, so there are 10 choices.

3. The tens digit is determined by the thousands digit, so there is only 1 choice.

4. The ones digit is determined by the hundreds digit, so there is only 1 choice.

To find the total number of palindromes, multiply the number of choices for each digit:

Total number of palindromes = (Number of choices for thousands digit) * (Number of choices for hundreds digit) * (Number of choices for tens digit) * (Number of choices for ones digit)

Total number of palindromes = 7 * 10 * 1 * 1 = 70

So, there are 70 integer palindromes between 10,000 and 70,000.

Next, let's find the number of even palindromes:

To be even, the ones digit must be even (i.e., 0, 2, 4, 6, or 8).

Since we already determined that the ones digit is determined by the hundreds digit, we need to identify the number of choices for the hundreds digit.

The hundreds digit can be any number from 0 to 9, so there are 10 choices.

To find the number of even palindromes, multiply the number of choices for each digit:

Total number of even palindromes = (Number of choices for thousands digit) * (Number of choices for hundreds digit) * (Number of choices for tens digit) * (Number of choices for ones digit)

Total number of even palindromes = 7 * 10 * 1 * 5 (since there are 5 even choices for the ones digit) = 350

Therefore, there are 350 even palindromes between 10,000 and 70,000.