To determine the number of 7-digit positive integers such that the sum of the digits is divisible by 3, we start by noting the characteristics of a 7-digit integer. A 7-digit integer takes the form \( d_1 d_2 d_3 d_4 d_5 d_6 d_7 \) where \( d_1 \) (the leading digit) is non-zero (ranging from 1 to 9) and \( d_2, d_3, d_4, d_5, d_6, d_7 \) can each range from 0 to 9.
Step 1: Total Number of 7-Digit Integers
The leading digit \( d_1 \) can be any digit from 1 to 9 (9 options), and each of the other 6 digits can be any digit from 0 to 9 (10 options each). Therefore, the total number of 7-digit integers can be calculated as follows:
\[ \text{Total 7-digit integers} = 9 \times 10^6 \]
Calculating this gives:
\[ 9 \times 10^6 = 9000000 \]
Step 2: Distribution of Sum of Digits
Next, we analyze the sum of the digits \( S = d_1 + d_2 + d_3 + d_4 + d_5 + d_6 + d_7 \). The sum \( S \) can take values ranging from a minimum (when \( d_1 = 1 \) and all other digits are 0, yielding \( S = 1 \)) to a maximum (when \( d_1 = 9 \) and all others are 9, yielding \( S = 9 + 6 \times 9 = 63 \)). Thus, \( S \) ranges from 1 to 63.
Step 3: Applying the Principle of Uniform Distribution
The key observation is that the sums of the digits will be uniformly distributed among the residues modulo 3. Since every digit can be in the set {0, 1, 2} modulo 3, and since the digits \( d_2, d_3, d_4, d_5, d_6, d_7 \) can take on any value which regularly distributes residues modulo 3, the distribution of the sums of \( d_1 + d_2 + \cdots + d_7 \) should lead to around one-third of the configurations resulting in a sum \( S \) that is divisible by 3.
Step 4: Counting the Numbers Whose Sums Are Divisible by 3
Given that the sum can produce residues of 0, 1, or 2 when taken modulo 3, we can conclude that approximately:
\[ \text{Number of 7-digit integers with } S \equiv 0 \mod 3 \approx \frac{1}{3} \text{ of total 7-digit integers} \]
Calculating this gives:
\[ \text{Count of } S \equiv 0 \mod 3 \approx \frac{9000000}{3} = 3000000 \]
Final Count
Thus, the final answer for the number of 7-digit positive integers where the sum of the digits is divisible by 3 is:
\[ \boxed{3000000} \]
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