Question 1
SEAMO self-study edition 2027; not an official 2027 paper. Original source years are retained. 25 questions, 90 minutes; Q1–10: 3 points, Q11–20: 4 points, Q21–25: 6 points, total 100. Original response types and all subparts are preserved. Verified is an editorial convention; source defects, necessary assumptions and unresolved mathematical uncertainty are disclosed.
$\overline{abc}$ is a three-digit number with no repeated digits. Given that $$\overline{ab}+\overline{ba}+\overline{ac}+\overline{ca}+\overline{bc}+\overline{cb}=\overline{abc},$$ the sum of the digits of $\overline{abc}$ is a multiple of ____. A bar denotes a number formed by the indicated digits.
The source does not ask for the greatest divisor, so more than one answer may fit the blank. Give any positive integer that divides every possible digit sum.
$\overline{abc}$ is a three-digit number with no repeated digits. Given that $$\overline{ab}+\overline{ba}+\overline{ac}+\overline{ca}+\overline{bc}+\overline{cb}=\overline{abc},$$ the sum of the digits of $\overline{abc}$ is a multiple of ____. A bar denotes a number formed by the indicated digits.
The source does not ask for the greatest divisor, so more than one answer may fit the blank. Give any positive integer that divides every possible digit sum.