Consecutive Integers — Practice Set

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

Consecutive integers are evenly spaced, so mean equals median, the sum is count x (first + last)/2, and the product of k consecutive integers is always divisible by k!.

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-CON-001 · MCQ · 40s

How many integers are there from -8 to 12, inclusive?

(A) 4
(B) 19
(C) 20
(D) 21
(E) 22

Q2 · NUM-CON-002 · MCQ · 45s

The largest of three consecutive integers is 17. What is the sum of the three integers?

(A) 45
(B) 48
(C) 51
(D) 54
(E) 57

Q3 · NUM-CON-007 · Numeric Entry · 55s

What is the sum of all integers from -6 to 10, inclusive? Enter your answer as a number.

Numeric entry — write the number.

Q4 · NUM-CON-008 · MCQ · 50s

Five consecutive odd integers have a sum of 85. What is the largest of these integers?

(A) 17
(B) 19
(C) 21
(D) 23
(E) 25

Q5 · NUM-CON-009 · QC · 45s

n is a positive integer.

Column A: The sum of n, n+1, and n+2 Column B: 3n + 3

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Level 2 · Medium — 8 questions · ~10 min

two or three steps, one planted trap each

Q6 · NUM-CON-014 · MCQ · 80s

The sum of 6 consecutive integers is 129. What is the sum of the largest and smallest of these integers?

(A) 19
(B) 24
(C) 41
(D) 42
(E) 43

Q7 · NUM-CON-017 · MCQ · 70s

The product of two consecutive positive integers is 156. What is their sum?

(A) 25
(B) 26
(C) 27
(D) 32
(E) 79

Q8 · NUM-CON-018 · Select all that apply · 85s

Which of the following MUST be true for any set of five consecutive integers? Select all that apply.

(A) Their sum is divisible by 5
(B) Their sum is divisible by 10
(C) Their product is divisible by 120
(D) Exactly one of the five integers is divisible by 5
(E) At least one of the five integers is divisible by 6

Q9 · NUM-CON-023 · QC · 85s

p and q are consecutive integers with p < q.

Column A: p^2 + q Column B: q^2 + p

Q10 · NUM-CON-026 · MCQ · 75s

The sum of three consecutive odd integers is S. If the largest of the three integers is k, which of the following expresses S in terms of k?

(A) 3k - 6
(B) 3k - 3
(C) 3k
(D) 3k + 3
(E) 3k + 6

Q11 · NUM-CON-028 · Numeric Entry · 70s

The sum of five consecutive integers is 0. What is the product of these five integers? Enter your answer as a number.

Numeric entry — write the number.

Q12 · NUM-CON-031 · MCQ · 85s

The sum of four consecutive integers is greater than 54 and less than 74. How many such sets of four consecutive integers are there?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 19

Q13 · NUM-CON-033 · Select all that apply · 75s

Which of the following can be the sum of two consecutive odd integers? Select all that apply.

(A) 18
(B) 24
(C) 30
(D) 36
(E) 40

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Level 3 · Hard — 12 questions · ~21 min

where 162+ is won or lost

Q14 · NUM-CON-019 · Numeric Entry · 100s

The sum of the consecutive integers from 50 to n, inclusive, is 3825, where n > 50. What is n? Enter your answer as a number.

Numeric entry — write the number.

Q15 · NUM-CON-024 · Numeric Entry · 110s

What is the sum of all the integers from 100 to 600, inclusive, that are a multiple of 11, a multiple of 9, or both? Enter your answer as a number.

Numeric entry — write the number.

Q16 · NUM-CON-032 · Numeric Entry · 100s

The sum of all the integers from -25 to n, inclusive, is 53, where n > 0. What is n? Enter your answer as a number.

Numeric entry — write the number.

Q17 · NUM-CON-034 · MCQ · 105s

The product of four consecutive positive integers is 116,280. What is the sum of the four integers?

(A) 70
(B) 72
(C) 74
(D) 78
(E) 80

Q18 · NUM-CON-035 · QC · 105s

n is an integer greater than 1.

Column A: The remainder when the product n(n+1)(n+2)...(2n-1) of n consecutive integers is divided by (n+1)! Column B: 0

Q19 · NUM-CON-037 · MCQ · 105s

The sum of the first n positive integers is 120. What is the sum of the first 2n positive integers?

(A) 240
(B) 435
(C) 465
(D) 480
(E) 496

Q20 · NUM-CON-038 · Select all that apply · 100s

Which of the following statements are ALWAYS true for any set of 5 consecutive integers? Select all that apply.

(A) The sum is divisible by 5
(B) The product is divisible by 5
(C) The product is divisible by 120
(D) The sum of the squares is odd
(E) The median equals the mean

Q21 · NUM-CON-039 · QC · 100s

a, b and c are consecutive integers with a < b < c.

Column A: a^2 + b^2 + c^2 Column B: ((a + b + c)^2)/3

Q22 · NUM-CON-041 · MCQ · 110s

m and n are integers with 0 < m < n. The sum of all the integers from m to n, inclusive, is 245, and the sum of all the integers from m to n + 3, inclusive, is 338. What is the value of m?

(A) -19
(B) 19
(C) 20
(D) 24
(E) 29

Q23 · NUM-CON-043 · Numeric Entry · 105s

The sum of all the odd integers from 1 to k, inclusive, is 3025, where k is a positive odd integer. What is the sum of all the odd integers from k + 2 to 199, inclusive? Enter your answer as a number.

Numeric entry — write the number.

Q24 · NUM-CON-044 · MCQ · 115s

Set S contains all the integers from 1 to 100, inclusive. Set T contains all the integers from d + 1 to d + 100, inclusive, where d is a nonzero integer. The sum of the members of T is k times the sum of the members of S, where k is an integer. What is the greatest possible negative value of d?

(A) -202
(B) -101
(C) -100
(D) -1
(E) 101

Q25 · NUM-CON-045 · Select all that apply · 110s

Which of the following integers can be written as the sum of two or more consecutive positive integers? Select all that apply.

(A) 15
(B) 16
(C) 21
(D) 30
(E) 32

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Level 4 · Extreme — 5 questions · ~11 min

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

Q26 · NUM-CON-046 · MCQ · 125s

How many different sets of 3 or more consecutive positive integers have a sum of 360?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 12

Q27 · NUM-CON-047 · QC · 130s

n is an integer greater than 1. S is the sum of the integers from 1 to n^2, inclusive. T is the sum of the integers from n+1 to n^2, inclusive.

Column A: S/T Column B: (n^2 + 1)/(n^2 - n)

Q28 · NUM-CON-048 · Numeric Entry · 130s

A set of n consecutive integers, where n > 1, has a sum of 2016. What is the greatest possible value of n? Enter your answer as a number.

Numeric entry — write the number.

Q29 · NUM-CON-049 · Select all that apply · 125s

A set of consecutive integers has a mean of 15 and a sum greater than 250 and less than 350. Which of the following could be the number of integers in the set? Select all that apply.

(A) 16
(B) 17
(C) 18
(D) 22
(E) 23

Q30 · NUM-CON-050 · QC · 125s

k is an integer greater than 1. P is the product of all the integers from 1 to 2k, inclusive. Q is the product of all the even integers from 2 to 2k, inclusive.

Column A: P/Q Column B: k!

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