Clock arithmetic: remainders and modular arithmetic

A long-distance bus leaves Nairobi at 17:00 and the journey takes 30 hours. On the 24-hour clock, at what hour does it arrive? (Give a number from 0 to 23.)

Today is Tuesday. What day of the week will it be in 45 days?

Without multiplying the numbers out, find $(53 \cdot 86) \bmod 9$.

What is $-23 \bmod 6$? (Remember that a remainder modulo $6$ is one of $0, 1, \dots, 5$.)

Which of these remainders have a multiplicative partner modulo $10$? (Choose all that apply.)

Find the partner of $3$ modulo $11$: the number $b$ from $0$ to $10$ with $3b \bmod 11 = 1$.

Amara claims: “If $a \cdot b \bmod n = 0$, then $a \bmod n = 0$ or $b \bmod n = 0$.” For which modulus does a counterexample exist?

What is the last digit of $3^{2026}$?

In Lean, a % n is the remainder of a on division by n. Which statement says that the remainder of a product depends only on the remainders of the factors (Theorem 2)?

Write a function partners(n) that returns the list of remainders a in range(n) that have a multiplicative partner modulo n — some b in range(n) with a * b % n == 1. Then use it to compare partners(12) with partners(13).