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Number Theory and Integer Properties Flashcards

7 cards from real GRE practice questions. Tap to flip, then mark Knew It or Still Learning — missed cards come back until you master them.

Read the first 7 Number Theory and Integer Properties flashcards as text

  1. If x and y are consecutive integers, which of the following must be even?

    Answer: x · y

    Among any two consecutive integers one is even, so their product x·y is always even.

  2. What is the remainder when 2^30 is divided by 7?

    Answer: 1

    Powers of 2 mod 7 cycle 2,4,1 with period 3, and 30 is divisible by 3, giving remainder 1.

  3. The product of three consecutive integers is always divisible by which number?

    Answer: 6

    Three consecutive integers always include a multiple of 2 and a multiple of 3, so the product is divisible by 6.

  4. If the GCD of two numbers is 1, the numbers are called what?

    Answer: Relatively prime

    Two integers with a greatest common divisor of 1 are relatively prime (coprime).

  5. How many positive integers less than 30 are relatively prime to 30?

    Answer: 8

    Using Euler's totient, φ(30) = 30(1−1/2)(1−1/3)(1−1/5) = 8.

  6. If n² is divisible by 12, what is the smallest positive integer value of n?

    Answer: 6

    For n² to contain 2²·3, n must contain a factor of 2 and 3, so the smallest n is 6.

  7. What is the remainder when the sum 1 + 2 + 3 + ... + 50 is divided by 7?

    Answer: 1

    The sum is 50·51/2 = 1275, and 1275 = 7·182 + 1, leaving remainder 1.