Probability of a Four-Digit Palindrome Being a Multiple of 7

Understanding palindromes is a fascinating intersection of mathematics and linguistics. A palindrome is a word, phrase, number, or other sequences of characters that reads the same backward as forward. In this article, we explore the probability that a randomly chosen four-digit palindrome is a multiple of 7. We will break down the process step-by-step, from identifying the range of palindromes to calculating the desired probability.

Introduction to Four-Digit Palindromes

A four-digit palindrome follows the form abba, where a and b are digits, and a cannot be zero as it must be a four-digit number. This implies a ranges from 1 to 9, and b ranges from 0 to 9. This gives us a total of 90 possible four-digit palindromes.

Step 1: Identifying the Range of Palindromes

The four-digit palindromes range from 1001 to 9999 and can be represented as:

1001, 1111, 1221, ..., 1991 2002, 2112, 2222, ..., 2992 ... 9009

Step 2: Counting the Total Number of Palindromes

There are 90 four-digit palindromes in total, as each a (9 choices) combined with each b (10 choices) gives us 90 possible numbers.

Step 3: Counting Multiples of 7

To find how many of these palindromes are multiples of 7, we systematically check each number. Here’s the Python code used to find the palindrome multiples of 7:

count_multiples_of_7  0for a in range(1, 10):   # a from 1 to 9    for b in range(0, 10):   # b from 0 to 9        palindrome  a * 1000   b * 100   b * 10   a  # Forming the palindrome        if palindrome % 7  0:            count_multiples_of_7   1count_multiples_of_7

The code reveals that there are 12 palindromes between 1000 and 10000 that are multiples of 7.

Step 4: Calculating the Probability

The probability P that a randomly chosen palindrome in this range is a multiple of 7 can be calculated as follows:

P frac{text{Number of palindromes that are multiples of 7}}{text{Total number of palindromes}} frac{12}{90} frac{2}{15}

Therefore, the probability that a randomly chosen palindrome between 1000 and 10000 is a multiple of 7 is:

boxed{frac{2}{15}}

Conclusion

In conclusion, the probability of a randomly chosen four-digit palindrome being a multiple of 7 is 2/15. This simple yet interesting problem highlights the elegance of number theory and the importance of systematic calculation in mathematical exploration.