Even & Odd Parity — Practice Set

Bank code: NUM-EOP · Section: Number Properties · 30 questions — Easy 5 · Medium 8 · Hard 12 · Extreme 5

Parity questions are classification questions, not calculation questions: odd + odd = even, even x anything = even, zero is even, and division has no parity rule.

Every question below is live in the question bank under the ID shown — the sheet and the portal are the same questions. Attempt a level with the clock running, then check Part C.

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Part A — Questions

Quantitative Comparison — the four choices are always the same, so they are not reprinted: (A) Column A is greater · (B) Column B is greater · (C) The two quantities are equal · (D) The relationship cannot be determined from the information given

Level 1 · Easy — 5 questions · ~4 min

warm-up — these must be automatic

Q1 · NUM-EOP-002 · MCQ · 45s

p is an even integer and q is an odd integer. Which of the following must be even?

(A) p + q
(B) p x q
(C) p + p + q
(D) q + q + q
(E) p x q + q

Q2 · NUM-EOP-004 · Numeric Entry · 40s

x is an even integer and y is an odd integer. What is the remainder when x + y is divided by 2? Enter your answer as a number.

Numeric entry — write the number.

Q3 · NUM-EOP-008 · Numeric Entry · 45s

How many integers from -6 to 7, inclusive, are even? Enter your answer as a number.

Numeric entry — write the number.

Q4 · NUM-EOP-010 · QC · 55s

a is an even integer and b is an odd integer.

Column A: a + b Column B: a x b

Q5 · NUM-EOP-011 · Select all that apply · 55s

n is an integer. For which of the following values of n is n^3 - n^2 + n even? Select all that apply.

(A) n = 3
(B) n = 4
(C) n = -2
(D) n = 0
(E) n = 7

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Level 2 · Medium — 8 questions · ~10 min

two or three steps, one planted trap each

Q6 · NUM-EOP-014 · MCQ · 70s

a is an even integer and b is an odd integer. Which of the following must be even?

(A) a^2 x b + b
(B) a x b + b
(C) a^2 x b + a
(D) a x b + b + a
(E) a^2 x b^2 + b

Q7 · NUM-EOP-017 · MCQ · 80s

m and n are integers such that m - n is even and m x n is even. Which of the following must be true?

(A) m + n is divisible by 4
(B) m x n is divisible by 4
(C) m and n are both odd
(D) m + n is odd
(E) m/2 + n/2 is an even integer

Q8 · NUM-EOP-021 · MCQ · 80s

j is an odd integer and k is an even integer. Which of the following must be odd?

(A) j x k
(B) j + k + 1
(C) j^2 + k^2
(D) j x k + k
(E) (j + k)(j - k) + 1

Q9 · NUM-EOP-022 · Numeric Entry · 70s

a and b are positive integers, a is even and b is odd. If a + b = 15 and a x b = 50, what is the value of a? Enter your answer as a number.

Numeric entry — write the number.

Q10 · NUM-EOP-024 · QC · 90s

n is a positive integer.

Column A: The remainder when n(n + 1)(n + 2) is divided by 4 Column B: 0

Q11 · NUM-EOP-031 · Numeric Entry · 65s

For how many integers n with -4 <= n <= 5 is 6n + 3 an odd integer? Enter your answer as a number.

Numeric entry — write the number.

Q12 · NUM-EOP-032 · QC · 65s

p is a positive even integer and q is a positive odd integer.

Column A: (-1)^p Column B: (-1)^q

Q13 · NUM-EOP-033 · MCQ · 75s

The product of two positive integers is 36, and one of the two integers is odd. What is the greatest possible value of the other integer?

(A) 6
(B) 9
(C) 12
(D) 18
(E) 36

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Level 3 · Hard — 12 questions · ~20 min

where 162+ is won or lost

Q14 · NUM-EOP-025 · MCQ · 95s

p is a positive even integer and q is a positive odd integer. Which of the following must be even?

(A) p^q - q^p
(B) (p + q)^q + 1
(C) p^q + q^q
(D) q^(p + q)
(E) (pq)^0 + p

Q15 · NUM-EOP-029 · MCQ · 95s

n is an integer. Which of the following expressions is odd for every integer n?

(A) n^3 + n
(B) n^3 + n^2
(C) n^3 + n + 1
(D) n^3 - n
(E) n^3 + n^2 + n + 1

Q16 · NUM-EOP-034 · MCQ · 105s

w, x, y and z are integers such that x + y is odd, y + z is odd and z + w is odd. Which of the following must be odd?

(A) x + z
(B) x + w
(C) w + y
(D) xy + zw
(E) x + yz

Q17 · NUM-EOP-035 · QC · 90s

m and n are positive integers with m < n. The product mn is even and the sum m + n is odd.

Column A: m^2 n Column B: m n^2

Q18 · NUM-EOP-036 · Select all that apply · 95s

a, b and c are positive integers and a^2 x b^3 x c is odd. Which of the following must be true? Select all that apply.

(A) a is odd
(B) b + c is even
(C) a + b + c is odd
(D) a x b x c + 1 is odd
(E) a^b is even

Q19 · NUM-EOP-037 · Numeric Entry · 95s

n is a positive integer less than 500. n/2 is an even integer, n/4 is an even integer, and n/8 is an odd integer. What is the greatest possible value of n? Enter your answer as a number.

Numeric entry — write the number.

Q20 · NUM-EOP-038 · MCQ · 100s

p and q are distinct prime numbers and p + q is odd. Which of the following must be true?

(A) p + q is prime
(B) p x q + 1 is odd
(C) p^2 + q^2 is even
(D) p + q > 7
(E) p x q is a perfect square

Q21 · NUM-EOP-041 · Select all that apply · 105s

x is an integer and x^3 + x is divisible by 4. Which of the following must be true? Select all that apply.

(A) x is even
(B) x is a multiple of 4
(C) x^2 + x is divisible by 8
(D) x/2 is an even integer
(E) x^3 + x is divisible by 8

Q22 · NUM-EOP-043 · MCQ · 105s

a, b and c are non-negative integers and 2^a x 3^b + 2^c is odd. Which of the following must be true?

(A) a = 0
(B) b = 0
(C) a + c is at least 1
(D) a = c
(E) b is even

Q23 · NUM-EOP-044 · Numeric Entry · 110s

a, b, c and d are integers, each greater than 1, and a x b x c x d = 2520. Exactly one of the four integers is even. What is the greatest possible value of that even integer? Enter your answer as a number.

Numeric entry — write the number.

Q24 · NUM-EOP-045 · QC · 105s

k is a number such that k^2 + k is an odd integer.

Column A: k Column B: 0

Q25 · NUM-EOP-051 · MCQ · 110s

The sum of seven integers is odd and the product of the same seven integers is even. Which of the following must be true?

(A) At least two of the seven integers are even
(B) At least four of the seven integers are odd
(C) All seven integers are odd
(D) Exactly one of the seven integers is even
(E) The product of the seven integers is divisible by 8

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Level 4 · Extreme — 5 questions · ~10 min

165+ — expect to need the insight, not the grind

Q26 · NUM-EOP-046 · Select all that apply · 125s

For a positive integer n, let f(n) be the sum of all positive divisors of n. For which of the following values of n is f(n) odd? Select all that apply.

(A) n = 6
(B) n = 8
(C) n = 12
(D) n = 18
(E) n = 25

Q27 · NUM-EOP-047 · MCQ · 120s

a, b and c are integers such that a + b + c = 0 and ab + bc + ca = -7. What can be said about a^2 + b^2 + c^2?

(A) It is always even
(B) It is always odd
(C) It could be either even or odd
(D) It is always divisible by 4
(E) It is always divisible by 3

Q28 · NUM-EOP-048 · QC · 125s

n is a positive integer. P(n) is the product of all odd integers from 1 to 2n - 1, inclusive, and S(n) is the sum of all integers from 1 to 2n, inclusive.

Column A: The remainder when P(n) is divided by 2 Column B: The remainder when S(n) is divided by 2

Q29 · NUM-EOP-049 · Numeric Entry · 130s

For how many integers n with 1 <= n <= 200 is n(n + 1)(n + 2) divisible by 8? Enter your answer as a number.

Numeric entry — write the number.

Q30 · NUM-EOP-050 · Select all that apply · 130s

For a positive integer k, let E(k) be the largest power of 2 that divides k. For example, E(12) = 4, because 12 = 4 x 3 and 8 does not divide 12. Which of the following statements are true for all positive integers n and m? Select all that apply.

(A) E(2n) = 2 x E(n)
(B) E(3n) = 3 x E(n)
(C) If n is odd, then E(n) = 1
(D) E(n^2) = [E(n)]^2
(E) E(n + m) = E(n) + E(m)

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