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
If n is a positive integer, what is the remainder when 7^100 is divided by 5?
Answer: 1
Powers of 7 mod 5 cycle 2,4,3,1 with period 4, and 100 is divisible by 4, giving remainder 1.
How many positive divisors does 72 have?
Answer: 12
Since 72 = 2^3 × 3^2, the number of divisors is (3+1)(2+1) = 12.
What is the greatest common divisor (GCD) of 84 and 126?
Answer: 42
84 = 2²·3·7 and 126 = 2·3²·7, so the GCD is 2·3·7 = 42.
If the integer k leaves remainder 3 when divided by 7, what is the remainder when 2k is divided by 7?
Answer: 6
If k ≡ 3 (mod 7), then 2k ≡ 6 (mod 7), giving remainder 6.
Which of the following is a prime number?
Answer: 83
83 has no divisors other than 1 and itself, while 91=7×13, 87=3×29, and 85=5×17.
What is the least common multiple (LCM) of 12 and 18?
Answer: 36
12 = 2²·3 and 18 = 2·3², so the LCM is 2²·3² = 36.
If n is an even integer, which expression must always be odd?
Answer: n² + 1
An even number squared is even, so n² + 1 is always odd.