There are 18 four-digit palindromic numbers divisible by 7.
Understanding Four-Digit Palindromes
A four-digit palindrome has the form abba, where a is a digit from 1 to 9 (since the number must be four digits) and b is a digit from 0 to 9. We can express this number as:
1000a + 100b + 10b + a = 1001a + 110b
Divisibility by 7
For the palindrome to be divisible by 7, the expression 1001a + 110b must be divisible by 7. We know that 1001 is divisible by 7 (1001 = 7 143). Therefore, 1001a is always divisible by 7, regardless of the value of a. So we just need to find the values of b that make 110b* divisible by 7.
110b can be written as (7 15 + 5)b = 7 15b + 5b. So, we need to find values of b such that 5b is divisible by 7. This is the same as finding values of b such that b is divisible by 7 because 5 and 7 are relatively prime.
Therefore, b must be a multiple of 7. Since b is a digit (0-9), the possible values for b are 0 and 7.
Counting the Palindromes
- When b = 0, the palindrome is of the form a00a. a can be any digit from 1 to 9, giving us 9 palindromes: 1001, 2002, 3003, 4004, 5005, 6006, 7007, 8008, 9009.
- When b = 7, the palindrome is of the form a77a. a can be any digit from 1 to 9, giving us 9 palindromes: 1771, 2772, 3773, 4774, 5775, 6776, 7777, 8778, 9779.
In total, there are 9 + 9 = 18 such palindromes.
