Divisibility (AIME / USAJMO / AMO)

Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.

Proof track: treat each problem as a full olympiad writeup on paper. Here you only enter the final answer (integer or short expression).

Divisibility: core idea

Valuations.

**(Proof track.)** Give a complete justification. Prove the digit criterion applies and find the smallest $a$ with $8a37$ divisible by $9$ .

Hint: State the congruence or counting principle used at each step. Answer: 0.

No proof submission required — answers only.

1. **(Proof track.)** Give a complete justification. Find the least positive $n$ for which $17\mid 2^n-1$ .
2. **(Proof track.)** Give a complete justification. Find the number of multiples of $12$ or $18$ in $[1,1000]$ .
3. **(Proof track.)** Give a complete justification. Find the remainder of $10^{100}+7^{100}$ upon division by $13$ .
4. **(Proof track.)** Give a complete justification. How many digits $a$ make $a428$ divisible by $11$ ?
5. **(Proof track.)** Give a complete justification. Find the least positive multiple of $27$ whose digit sum is $9$ .
6. **(Proof track.)** Give a complete justification. Find $n\pmod {35}$ if $n\equiv4\pmod5,n\equiv3\pmod7$ .
7. **(Proof track.)** Give a complete justification. Find the last digit of $7^{2026}$ .
8. **(Proof track.)** Give a complete justification. Find the remainder of $2^{50}+3^{50}$ modulo $7$ .
9. **(Proof track.)** Give a complete justification. How many integers below $1000$ are relatively prime to $30$ ?