Understanding the Divisibility of 4-Digit Numbers

A number is divisible by 3 if and only if the sum of its digits is divisible by 3. This rule is crucial in determining which 4-digit numbers can be formed using the digits 1 through 6 with repetition allowed, and which of these are divisible by 3.

Identifying Multiples of 3

The multiples of 3 that can arise from combinations of the digits 1 through 6 are 6, 9, 12, 15, 18, 21, and 24. We need to find out how many 4-digit numbers can be formed that fit this criterion, given the condition of repetition.

Calculating Total 4-Digit Numbers

The total number of 4-digit natural numbers that can be formed using the digits 1 through 6, with repetition allowed, is given by (6 times 6 times 6 times 6 1296). This is because each digit position has 6 possible choices.

Counting Multiples of 3

It has been demonstrated that exactly (1296/3 432) of these numbers are divisible by 3. This result is derived by arranging the 1296 numbers in increasing order and noting that in each set of six consecutive numbers, exactly two are multiples of 3.

For a more in-depth understanding, let's examine how these numbers are arranged and counted:

The list of all 4-digit numbers from 1111 to 6666 is divided into rows, each containing six consecutive numbers. In each of these rows, two numbers are multiples of 3. Here is a partial list to illustrate this:

1111 1112 1113 1114 1115 1116 1121 1122 1123 1124 1125 1126 1131 1132 1133 1134 1135 1136 ... (continues up to 6666) ... (continues up to 6666) ... (continues up to 6666) ... (continues up to 6666) ... (continues up to 6666) ... (continues up to 6666)

Note that each row contains six consecutive numbers, and there are exactly two multiples of 3 in each row. Since the entire list contains 216 such rows, the total count of multiples of 3 is (216 times 2 432), which confirms the previous calculation.

Conclusion

In conclusion, the number of 4-digit numbers that can be formed using the digits 1 through 6 with repetition allowed, and that are divisible by 3, is 432. This result is derived through systematic analysis and confirmation by examining the arrangement of these numbers.