To solve the problem, we analyze the conditions for a four-digit number (N = 1000a + 100b + 10c + d) and its reverse (M = 1000d + 100c + 10b + a) such that (N + M) is divisible by 12.

Key Observations:

• (N + M = 1001(a+d) + 110(b+c))
• For (N + M) to be divisible by 12, the expression (7(a+d) + 10(b+c) \equiv 0 \mod 12) (simplified using modular arithmetic).
• This reduces to (S_2 \equiv 2S_1 \mod 12) where (S_1 = a+d) and (S_2 = b+c).

Counting Valid Pairs:

• (a) (1-9) and (d) (0-9): For each (S_1), the number of valid ((a,d)) pairs is calculated.
• (b) and (c) (0-9): For each residue of (S_2 \mod 12), the number of valid ((b,c)) pairs is consistent.

Summing these valid pairs across all possible values gives the total number of such four-digit numbers.

Conclusion:

The total number of valid four-digit numbers is 1080.

Answer: (\boxed{1080})

作者声明:本文包含人工智能生成内容。