Digit-repetition product — Find the smallest positive integer N such that 13 * N equals a number consisting entirely of the digit 5 (for example, 55, 555, 5555, …). Determine the minimal N that works.

Introduction / Context:
This number-system puzzle asks for the smallest positive integer N such that 13 * N produces a repdigit made only of 5s (like 5, 55, 555, 5555, …). Problems of this type are efficiently handled with modular arithmetic and repetition-number (repunit) reasoning, avoiding brute-force search.

Given Data / Assumptions:

• We need 13 * N = 55…5 (k copies of 5) for some smallest integer k.
• The target number with k copies of 5 equals 5 * (10^k - 1) / 9.
• N must be a positive integer; k must be minimal to ensure N is minimal.

Concept / Approach:
Let T_k = 5 * (10^k - 1) / 9. We need 13 | T_k. Since gcd(5, 13) = 1 and gcd(9, 13) = 1, this reduces to 13 | (10^k - 1). Thus, the smallest viable k is the multiplicative order of 10 modulo 13 (or modulo 117 when considering the 9 in the denominator rigorously). The minimal k that makes (10^k - 1) divisible by 117 yields the first all-5s number divisible by 13.

Step-by-Step Solution:
Write the repdigit: 55…5 (k fives) = 5 * (10^k - 1) / 9.We need 13 | 5 * (10^k - 1) / 9. Since 5 and 13 are coprime, ensure 117 | (10^k - 1).The smallest k with 10^k ≡ 1 (mod 117) is k = 6 (standard order result).Thus, the first suitable repdigit is 555555. Compute N = 555555 / 13 = 42735.

Verification / Alternative check:
Check directly: 13 * 42735 = 555555. No shorter repdigit of 5s (like 5, 55, 555, 5555, 55555) is divisible by 13, so this N is minimal.

Why Other Options Are Wrong:

• 41625 / 42515 / 42135 / 55555: None of these, when multiplied by 13, yield an all-5s product. Only 42735 satisfies 13 * N = 555555 exactly.

Common Pitfalls:
Trying random guesses; forgetting the repdigit formula 5 * (10^k - 1) / 9; checking divisibility by 13 without confirming minimality of k.

Final Answer:
42735