<p>In the decimal system of numeration of six-digit numbers in which the sum of the digits is divisible by 5 is</p>
Step-by-Step Solution
Key Concept: Use generating functions or digit constraint: for six-digit numbers (first digit 1-9, others 0-9), count arrangements where digit sum ≡ 0 (mod 5). By symmetry, exactly 1/5 of all valid numbers satisfy this condition.
<p><strong>Step 1:</strong> Count total six-digit numbers: First digit has 9 choices (1-9), remaining 5 digits have 10 choices each = 9 × 10⁵</p><p><strong>Step 2:</strong> For digit sum divisibility by 5, use symmetry principle. Fix the first 5 digits arbitrarily. The 6th digit must satisfy: (sum of first 5 digits + d₆) ≡ 0 (mod 5), where d₆ ∈ {0,1,...,9}</p><p><strong>Step 3:</strong> For any fixed sum of first 5 digits, exactly 2 values of d₆ from {0,1,...,9} satisfy the congruence (since 10 ≡ 0 (mod 5), the remainders cycle: {0,1,2,3,4,0,1,2,3,4})</p><p><strong>Step 4:</strong> Total count = 9 × 10⁴ × 2 = <strong>1,80,000</strong></p><p>∴ Answer: A</p>
Correct Answer: A