Divisibility Criteria

To determine if a number is divisible by 42, it must also be divisible by 2, 3, and 7. Here are the rules for each:

• Divisibility by 2: A number qualifies as divisible by 2 if it ends in an even digit (0, 2, 4, 6, or 8).
• Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3. For instance, 927 is divisible by 3 because 9 + 2 + 7 = 18, which is divisible by 3.
• Divisibility by 7: For a number N to be divisible by 7, the difference between N (with its last digit removed) and twice the last digit must also be divisible by 7. For example, 434 is divisible by 7 since 43 - 8 = 35 is divisible by 7.

If you're curious about the rationale behind these rules, a quick online search will yield plenty of resources. For details specifically about the rule for 7, you can refer to my earlier article.

Section 1.1: Filtering Palindromic Numbers

For a 4-digit palindromic number to be divisible by 42, it must satisfy all three divisibility criteria mentioned above. First, we can eliminate palindromic numbers that start with 1, 3, 5, 7, or 9, since they will also end with those digits and thus cannot be even.

We can narrow our focus to 4-digit palindromic numbers that begin with 2, 4, 6, or 8, ensuring they are divisible by 2. Now, the question remains: which of these numbers are also divisible by 3 and 7, and consequently by 42?

Section 1.2: Palindromic Numbers of the Form abba

Let's first examine the numbers divisible by 7, as this will yield a smaller subset to work with. From this list, we will then filter out those that are not divisible by 3.

Considering palindromic numbers of the structure abba, where a = 2, 4, 6, or 8 and b = 0, 1, ..., 9 (e.g., 2112), we can apply our divisibility test for 7. A number abba is divisible by 7 if:

7 divides (100a + 10b + b) - 2a,

which simplifies to:

7 divides 98a + 11b,

or more succinctly:

7 divides 11b.

This condition is satisfied only when b = 7.

The remaining 4-digit palindromic numbers that are divisible by both 2 and 7 are 2772, 4774, 6776, and 8778. Among these, only 2772 and 8778 meet the criteria for divisibility by 3 as well.

To sum up, the only 4-digit palindromic numbers that are divisible by 42 are 2772 and 8778.