Remainders & Cyclicity — Practice Set

Bank code: NUM-REM · Section: Arithmetic · 30 questions — Easy 5 · Medium 8 · Hard 12 · Extreme 5

Write N = dq + r, reduce at every step instead of at the end, and remember that powers of a fixed base repeat with a period of 4 or less.

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-REM-004 · MCQ · 45s

When a positive integer n is divided by 6, the remainder is 0. Which of the following must be true?

I. n is divisible by 2

II. n is divisible by 3

III. n is divisible by 12

(A) I only
(B) II only
(C) I and II only
(D) I and III only
(E) I, II, and III

Q2 · NUM-REM-005 · Numeric Entry · 40s

When 53 is divided by 8, the remainder is r. What is the value of r? Enter your answer as a number.

Numeric entry — write the number.

Q3 · NUM-REM-006 · QC · 45s

n is a positive integer.

Column A: The remainder when 2n is divided by 2 Column B: The remainder when 2n + 1 is divided by 2

Q4 · NUM-REM-007 · MCQ · 40s

What is the units digit of 7 x 48?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 8

Q5 · NUM-REM-011 · MCQ · 45s

The positive integer n leaves a remainder of 4 when divided by 7. What is the remainder when n + 7 is divided by 7?

(A) 0
(B) 3
(C) 4
(D) 7
(E) 11

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Level 2 · Medium — 8 questions · ~10 min

two or three steps, one planted trap each

Q6 · NUM-REM-014 · MCQ · 70s

When the integer n is divided by 8, the remainder is 5. What is the remainder when 4n is divided by 8?

(A) 0
(B) 3
(C) 4
(D) 5
(E) 20

Q7 · NUM-REM-015 · MCQ · 70s

What is the remainder when 2^10 is divided by 5?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 0

Q8 · NUM-REM-017 · QC · 75s

When the integer m is divided by 6 the remainder is 4, and when the integer n is divided by 6 the remainder is 3.

Column A: The remainder when mn is divided by 6 Column B: The remainder when m + n is divided by 6

Q9 · NUM-REM-019 · Select all that apply · 75s

The positive integer n leaves a remainder of 5 when divided by 12. Which of the following could be the value of n? Select all that apply.

(A) 5
(B) 17
(C) 24
(D) 29
(E) 37

Q10 · NUM-REM-023 · MCQ · 70s

When the positive integer k is divided by 9, the remainder is 7. What is the remainder when k + 20 is divided by 9?

(A) 0
(B) 2
(C) 7
(D) 9
(E) 27

Q11 · NUM-REM-029 · MCQ · 75s

When the positive integer m is divided by 14, the remainder is 9. What is the remainder when m is divided by 7?

(A) 0
(B) 2
(C) 5
(D) 7
(E) 9

Q12 · NUM-REM-030 · Select all that apply · 85s

The positive integer n leaves a remainder of 3 when divided by 8. Which of the following must be true? Select all that apply.

(A) n is odd
(B) The remainder when n is divided by 4 is 3
(C) The remainder when n^2 is divided by 8 is 1
(D) The remainder when n is divided by 16 is 3
(E) The remainder when 3n is divided by 8 is 9

Q13 · NUM-REM-031 · Numeric Entry · 70s

What is the remainder when 50! + 3 is divided by 7?

(Note: 50! = 50 x 49 x 48 x ... x 1.) 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-REM-034 · MCQ · 90s

What is the remainder when 2^100 is divided by 7?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

Q15 · NUM-REM-035 · Numeric Entry · 100s

When the positive integer n is divided by 5 the remainder is 3, and when n is divided by 7 the remainder is 4. What is the remainder when n is divided by 35? Enter your answer as a number.

Numeric entry — write the number.

Q16 · NUM-REM-036 · MCQ · 95s

What is the remainder when 3^100 + 4^100 is divided by 5?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

Q17 · NUM-REM-037 · QC · 90s

n is a positive integer, and when n is divided by 3 the remainder is 2.

Column A: The remainder when n^2 + n is divided by 3 Column B: The remainder when n^2 - n is divided by 3

Q18 · NUM-REM-038 · Select all that apply · 105s

A positive integer n leaves the same remainder when divided by 4 as it leaves when divided by 6. Which of the following could be the value of n? Select all that apply.

(A) 14
(B) 20
(C) 25
(D) 30
(E) 39

Q19 · NUM-REM-039 · Numeric Entry · 110s

What is the remainder when 17^50 is divided by 100? Enter your answer as a number.

Numeric entry — write the number.

Q20 · NUM-REM-040 · QC · 95s

p is a prime number greater than 3.

Column A: The remainder when p^2 is divided by 12 Column B: 1

Q21 · NUM-REM-041 · MCQ · 90s

n is a positive integer. What is the remainder when 5^n + 5^(n+1) is divided by 6?

(A) 0
(B) 1
(C) 5
(D) 6
(E) The remainder depends on whether n is odd or even

Q22 · NUM-REM-042 · Numeric Entry · 100s

What is the remainder when 1! + 2! + 3! + ... + 20! is divided by 12? Enter your answer as a number.

Numeric entry — write the number.

Q23 · NUM-REM-043 · MCQ · 105s

For each positive integer n, let R(n) denote the remainder when n is divided by 11. What is the value of R(1) + R(2) + R(3) + ... + R(50)?

(A) 220
(B) 235
(C) 241
(D) 247
(E) 285

Q24 · NUM-REM-044 · QC · 90s

t is a positive integer.

Column A: The remainder when 6t + 5 is divided by 6 Column B: The remainder when 6t + 5 is divided by 5

Q25 · NUM-REM-045 · Select all that apply · 105s

Which of the following are equal to the remainder when 8^400 is divided by 5? Select all that apply.

(A) 0
(B) 1
(C) The remainder when 2^400 is divided by 5
(D) The remainder when 3^400 is divided by 5
(E) 4

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Level 4 · Extreme — 5 questions · ~10 min

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

Q26 · NUM-REM-046 · Numeric Entry · 125s

What is the remainder when 2^1 + 2^2 + 2^3 + ... + 2^62 is divided by 7? Enter your answer as a number.

Numeric entry — write the number.

Q27 · NUM-REM-047 · MCQ · 125s

A positive integer n leaves a remainder of 2 when divided by 3, a remainder of 3 when divided by 5, and a remainder of 4 when divided by 7. What is the smallest possible value of n?

(A) 53
(B) 68
(C) 83
(D) 158
(E) 173

Q28 · NUM-REM-048 · QC · 125s

n is a positive integer.

Column A: The remainder when n^3 - n is divided by 6 Column B: The remainder when 2^n + 1 is divided by 3

Q29 · NUM-REM-049 · Numeric Entry · 120s

What is the units digit of 7^(7^7)?

(The exponent is 7^7 = 823,543, so this is 7 raised to the power 823,543.) Enter your answer as a number.

Numeric entry — write the number.

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

a, b and d are positive integers with d > 1. Which of the following must be true? Select all that apply.

(A) If a leaves remainder r when divided by d, then 2a leaves remainder 2r when divided by d
(B) If a and b leave the same remainder when divided by d, then d divides a - b
(C) If a leaves remainder r when divided by d, then a leaves remainder r when divided by every factor of d that is greater than 1
(D) The remainder when a^2 is divided by d is completely determined by the remainder when a is divided by d
(E) If a is divisible by d, then a is divisible by every multiple of d

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