Number Theory Essentials — Practice Set

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

"Number" is not "integer": every must-be-true question has to be tested against fractions, zero and negatives before you trust it.

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-NTH-001 · MCQ · 45s

Which of the following is NOT a rational number?

(A) 0
(B) -7/4
(C) sqrt(10)
(D) 0.666...
(E) 3.25

Q2 · NUM-NTH-003 · MCQ · 55s

The integer 0 is a factor of the integer k. Which of the following must be true?

(A) k = 1
(B) k = 0
(C) k can be any integer, because 0 is a multiple of every integer
(D) k can be any even integer, because 0 is even
(E) No such integer k exists, because 0 is a factor of no non-zero integer

Q3 · NUM-NTH-004 · QC · 45s

Column A: The number of integers n with -2.5 < n < 3.5 Column B: 6

Q4 · NUM-NTH-005 · MCQ · 50s

Which of the following is true for every integer n?

(A) n^2 > 0
(B) n^2 >= 0
(C) n^2 > n
(D) n^2 is odd
(E) n^3 > 0

Q5 · NUM-NTH-006 · Numeric Entry · 55s

The repeating decimal 0.135135135... is equal to a/b, where a and b are positive integers with no common factor greater than 1. What is a + b? Enter your answer as a number.

Numeric entry — write the number.

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Level 2 · Medium — 8 questions · ~10 min

two or three steps, one planted trap each

Q6 · NUM-NTH-011 · MCQ · 70s

Which of the following numbers is irrational?

(A) 0.474747...
(B) sqrt(49)
(C) 22/7
(D) sqrt(45)
(E) -3/8

Q7 · NUM-NTH-012 · QC · 65s

n is an integer.

Column A: n^3 Column B: n^2

Q8 · NUM-NTH-013 · Numeric Entry · 75s

How many positive integers n less than 30 have the property that n^2 has units digit 4? Enter your answer as a number.

Numeric entry — write the number.

Q9 · NUM-NTH-014 · Select all that apply · 85s

n is an integer. Which of the following must be true? Select all that apply.

(A) n^3 >= n
(B) n^2 >= n
(C) n^3 >= n^2
(D) (n + 1)^2 > n^2
(E) |n^3| >= |n|

Q10 · NUM-NTH-015 · MCQ · 75s

a is a number with a > 1. Which of the following is the SMALLEST?

(A) a
(B) a^2
(C) sqrt(a)
(D) 1/a
(E) 1/a^2

Q11 · NUM-NTH-017 · MCQ · 80s

n is a positive integer that is not a perfect square. Which of the following must be true of sqrt(n)?

(A) sqrt(n) is irrational
(B) sqrt(n) is rational but not an integer
(C) sqrt(n) is an integer
(D) sqrt(n) is not a real number
(E) sqrt(n) may be rational or irrational, depending on n

Q12 · NUM-NTH-022 · Select all that apply · 90s

Which of the following integers have a number of positive factors that is itself a perfect square? Select all that apply.

(A) 24
(B) 36
(C) 225
(D) 512
(E) 1,000

Q13 · NUM-NTH-025 · Numeric Entry · 80s

For how many integers n with -5 <= n <= 5 is n^3 < n? Enter your answer as a number.

Numeric entry — write the number.

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Level 3 · Hard — 12 questions · ~20 min

where 162+ is won or lost

Q14 · NUM-NTH-027 · MCQ · 105s

Which of the following integers CANNOT be written as the difference of the squares of two integers?

(A) 1,000
(B) 1,001
(C) 1,002
(D) 1,004
(E) 1,005

Q15 · NUM-NTH-028 · QC · 95s

a is a number with 0 < a < 1.

Column A: sqrt(a) Column B: a^2 + a

Q16 · NUM-NTH-029 · Numeric Entry · 100s

For how many integers n with 1 <= n <= 90 does the fraction n/90 have a terminating decimal expansion? Enter your answer as a number.

Numeric entry — write the number.

Q17 · NUM-NTH-030 · Select all that apply · 105s

a is a number (not necessarily an integer) with a^3 < a. Which of the following must be true? Select all that apply.

(A) a < 1
(B) a^2 < 1
(C) a is not an integer
(D) a^5 < a
(E) 1/a > a

Q18 · NUM-NTH-031 · MCQ · 105s

a is a number such that a^2 is rational. Which of the following must also be rational?

(A) a
(B) a^3
(C) a^4
(D) a + a^2
(E) sqrt(a)

Q19 · NUM-NTH-032 · QC · 95s

a and b are non-zero numbers with a x b > 0 and a > b.

Column A: 1/b Column B: 1/a

Q20 · NUM-NTH-033 · MCQ · 105s

For how many positive integers n with 1 <= n <= 75 do n^2 and n^5 have the same units digit?

(A) 15
(B) 28
(C) 29
(D) 30
(E) 32

Q21 · NUM-NTH-034 · Numeric Entry · 95s

a is a positive number such that a + 1/a = 5. What is the value of a^2 + 1/a^2? Enter your answer as a number.

Numeric entry — write the number.

Q22 · NUM-NTH-035 · Select all that apply · 110s

Which of the following statements must be true for all numbers a and b? Select all that apply.

(A) If a is irrational, then 1/a is irrational
(B) If a^2 is irrational, then a is irrational
(C) If a is irrational, then a^2 is irrational
(D) If a and b are irrational, then a + b is irrational
(E) If a is irrational and r is a non-zero rational, then a/r is irrational

Q23 · NUM-NTH-036 · QC · 100s

a is a number such that a^5 = a^3.

Column A: a^2 Column B: a^4

Q24 · NUM-NTH-041 · Numeric Entry · 110s

N is an integer greater than 10^6 that is both a perfect square and a perfect cube, and N has more than one distinct prime factor. What is the least possible value of N? Enter your answer as a number.

Numeric entry — write the number.

Q25 · NUM-NTH-042 · MCQ · 105s

n is an integer with -6 <= n <= 6. For how many of these values of n is the expression (n^4 - 13n^2 + 36)/(n^2 - 5n + 6) defined and not equal to 0?

(A) 6
(B) 7
(C) 8
(D) 9
(E) 11

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Level 4 · Extreme — 5 questions · ~11 min

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

Q26 · NUM-NTH-037 · MCQ · 125s

For how many integers n with 1 <= n <= 51 is 12^n x 75 both a perfect square and a multiple of 6^12?

(A) 20
(B) 21
(C) 23
(D) 26
(E) 46

Q27 · NUM-NTH-038 · Numeric Entry · 130s

How many integers n with 1 <= n <= 1,000 are neither a perfect square, nor a perfect cube, nor a perfect fifth power? Enter your answer as a number.

Numeric entry — write the number.

Q28 · NUM-NTH-039 · QC · 130s

a and b are positive numbers with a + b = 1 and a is not equal to b.

Column A: 1/a + 1/b Column B: 4

Q29 · NUM-NTH-040 · MCQ · 125s

a and b are non-zero numbers with a/b > 1. Which of the following must be true?

(A) a > b
(B) a - b > 0
(C) a^2 > b^2
(D) a > 0
(E) 1/a < 1/b

Q30 · NUM-NTH-043 · Select all that apply · 130s

a is a number with 0 < a < 1. Which of the following must be true? Select all that apply.

(A) 1/(1 - a) > 1
(B) sqrt(1 - a^2) > 1 - a
(C) a^2 + a^3 > a
(D) a^2 + 1/a^2 > 2
(E) a + 1/a < 2

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