For any four different natural numbers, the difference between at least two numbers is a multiple of 3. Try to explain the reason

Using drawer principle: first of all, we need to make clear such a rule: if the remainder of two natural numbers divided by 3 is the same, then the difference between the two natural numbers is a multiple of 3. And the remainder of any natural number divided by 3 is either 0, or 1, or 2. According to these three situations, natural numbers can be divided into three types, and these three types are

There are five different natural numbers. Among them, the sum of any three numbers is a multiple of 3, the sum of any four numbers is a multiple of 4, and the sum is the smallest. What are the five numbers

If the sum of any three numbers is a multiple of 3, they must be of the same class. They can all be numbers in the form of 3k, 3K + 1, 3K + 2
Similarly, if the sum of any four numbers is a multiple of 4, they must be of the same class. They can all be numbers in the form of 4K, 4K + 1,4k + 2,4k + 3
Various situations can be considered respectively: if the five numbers are of 3K and 4K types, they must be in the form of 12K; if the five numbers are 3K and 4K + 1, they must be in the form of 12K + 9; if the five numbers are in the form of 3K and 4K + 2, they must be in the form of 12K + 6; if the five numbers are in the form of 3K and 4K + 3, they must be in the form of 12K + 3
We can also discuss other cases in turn, and find that these five numbers are all of the same type, with 12K + m (M = 0,1,2,...) 11) all meet the requirements
If the natural number you are talking about contains 0, then the minimum sum should be in the form of 12K 0,12,24,36,48. If the natural number you are talking about does not contain 0, then the minimum sum should be in the form of 12K + 1 1 1,13,25,37,49

Choose five different natural numbers (not equal to 0) so that the sum of any three numbers is a multiple of 3. What are the five numbers?
Please list the calculation process

If the sum of any three numbers is a multiple of 3, the remainder of all numbers divided by 3 should be the same
Then these numbers are at least: 1,4,7,10,13
And: 1 + 4 + 7 + 10 + 13 = 35
A: the minimum sum of these five numbers is 35

There are five different natural numbers. The sum of any three of them is a multiple of 3,
The sum of any four numbers is a multiple of four. To make the sum of the five numbers as small as possible, what are the five numbers?

Consider that these five numbers are divided by 3 and 4 respectively, and the remainder is 1, [3,4] = 12. Then the minimum of these five numbers is 1, 13, 25, 37 and 49

For any four different natural numbers, the difference between at least two numbers is a multiple of 3. Try to explain the reason