Modular toolkit

Questions 1 to 5 of 7

1. 1. An AI tool wrote this line for a modular product. The variables are long long. ``cpp long long r = a * b % 1000000007LL; // a and b are near 1e18 `` A reviewer says it is wrong. What is the reason?

   The % operator must come before the * operator, so the order is wrongThe code is correct, because % always keeps the result below the modulus even after an overflowThe code is correct, but it is slow, so it should use a loop insteadThe product of two numbers near 1e18 is far above the 64-bit limit, so it overflows before the remainder is taken, and the printed value is not a remainder
2. 2. What is 6 / 7 modulo 1000000007, as a number from 0 to 1000000006?

   Type a number.

3. 3. Which statements about the inverse of a modulo m are true?

   Pick every answer that is true.

   It exists when gcd(a, m) = 1For a prime m, it exists for every a that is not a multiple of mIt exists for every a when m is 10It exists for every a below m
4. 4. About how many squarings does binary exponentiation do when the exponent is 10^18?

   About 30About 10^18 / 2About 60About 1000
5. 5. Name the prime modulus used in the lesson's worked examples, as a plain integer.

   Type your answer.