Digits & Place Value — Practice Set

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

A two-digit number is 10a + b, not ab: write the place-value expansion first and every digit puzzle becomes ordinary algebra with digits restricted to 0 through 9.

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-DPV-001 · MCQ · 40s

In the integer 4,728, what is the value of the digit in the hundreds place?

(A) 2
(B) 4
(C) 7
(D) 72
(E) 700

Q2 · NUM-DPV-003 · Numeric Entry · 40s

What is the sum of the digits of 3,456? Enter your answer as a number.

Numeric entry — write the number.

Q3 · NUM-DPV-004 · MCQ · 40s

How many two-digit positive integers are there?

(A) 89
(B) 90
(C) 91
(D) 99
(E) 100

Q4 · NUM-DPV-006 · QC · 45s

A two-digit positive integer has tens digit a and units digit b, where a > b.

Column A: The integer itself Column B: The number formed by reversing its two digits

Q5 · NUM-DPV-008 · MCQ · 45s

The tens digit of a two-digit positive integer is twice its units digit. If the units digit is 3, what is the integer?

(A) 18
(B) 23
(C) 36
(D) 53
(E) 63

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Level 2 · Medium — 8 questions · ~11 min

two or three steps, one planted trap each

Q6 · NUM-DPV-011 · MCQ · 75s

A two-digit positive integer is 36 more than the integer formed by reversing its digits. If the product of its digits is 12, what is the integer?

(A) 26
(B) 34
(C) 43
(D) 62
(E) 84

Q7 · NUM-DPV-016 · Select all that apply · 80s

A two-digit positive integer has tens digit t and units digit u. Which of the following statements must be true? Select all that apply.

(A) The integer is divisible by t + u
(B) The integer equals 10t + u
(C) t can equal 0
(D) Reversing the digits produces the number 10u + t
(E) The difference between the integer and the number formed by reversing its digits is divisible by 9

Q8 · NUM-DPV-017 · Numeric Entry · 85s

In a four-digit positive integer, the thousands digit is 5, the hundreds digit and the units digit are both equal to the digit d, and the tens digit is 3. If the integer is divisible by 9, what is the integer? Enter your answer as a number.

Numeric entry — write the number.

Q9 · NUM-DPV-019 · QC · 85s

Column A: The number of three-digit positive integers that contain at least one digit equal to 0 Column B: 180

Q10 · NUM-DPV-020 · MCQ · 75s

A three-digit positive integer has hundreds digit a, tens digit b and units digit c. If a + b + c = 15 and a = c, which of the following could be the integer?

(A) 483
(B) 555
(C) 582
(D) 672
(E) 693

Q11 · NUM-DPV-021 · Numeric Entry · 75s

How many three-digit positive integers have a units digit of 5? Enter your answer as a number.

Numeric entry — write the number.

Q12 · NUM-DPV-022 · Select all that apply · 80s

Which of the following products have a units digit of 2? Select all that apply.

(A) 45 x 18
(B) 14 x 23
(C) 36 x 27
(D) 63 x 29
(E) 19 x 8

Q13 · NUM-DPV-024 · MCQ · 85s

A three-digit positive integer has hundreds digit h, tens digit 3 and units digit u. If h = 2u and the three digits sum to 15, what is the integer?

(A) 438
(B) 633
(C) 735
(D) 834
(E) 843

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Level 3 · Hard — 12 questions · ~21 min

where 162+ is won or lost

Q14 · NUM-DPV-027 · MCQ · 100s

N is a two-digit positive integer whose tens digit and units digit are both non-zero. The positive difference between N and the integer formed by reversing the digits of N is 54, and N is a multiple of 7. What is N?

(A) 17
(B) 28
(C) 39
(D) 71
(E) 93

Q15 · NUM-DPV-028 · Numeric Entry · 105s

A two-digit positive integer is 12 greater than the product of its two digits. What is the sum of all such integers? Enter your answer as a number.

Numeric entry — write the number.

Q16 · NUM-DPV-029 · MCQ · 110s

For how many three-digit positive integers N is N equal to 16 times the sum of the digits of N?

(A) 2
(B) 3
(C) 4
(D) 7
(E) 21

Q17 · NUM-DPV-032 · Select all that apply · 105s

A four-digit positive integer has the form AABB, where A is both the thousands and the hundreds digit, B is both the tens and the units digit, and A is not 0. Which of the following must be true for every such integer? Select all that apply.

(A) AABB is divisible by 11
(B) AABB is divisible by 3
(C) AABB divided by 11 is a three-digit integer
(D) The sum of the digits of AABB is even
(E) AABB is divisible by 101

Q18 · NUM-DPV-035 · Numeric Entry · 120s

The product of the three digits of the positive integer N is 36. What is the sum of all possible values of N? Enter your answer as a number.

Numeric entry — write the number.

Q19 · NUM-DPV-041 · MCQ · 95s

The sum of a two-digit positive integer and the integer formed by reversing its digits is 143. How many two-digit integers have this property?

(A) 6
(B) 7
(C) 9
(D) 11
(E) 13

Q20 · NUM-DPV-042 · MCQ · 105s

A decimal number has the form 4.d7k, so the three digits after the decimal point are d, 7 and k in that order, where k is a non-zero digit and d is a digit. The value of the digit in the tenths place is 250 times the value of the digit in the thousandths place. What is the sum of the values of the digits in the tenths and thousandths places?

(A) 0.052
(B) 0.07
(C) 0.502
(D) 0.52
(E) 7

Q21 · NUM-DPV-043 · QC · 95s

N is a two-digit positive integer.

Column A: The sum of the digits of the integer N + 10 Column B: The sum of the digits of N

Q22 · NUM-DPV-044 · Numeric Entry · 110s

N is a three-digit positive integer and M is the integer formed by writing the digits of N in reverse order. If N - M = 297 and the digits of N sum to 18, how many possible values of N are there? Enter your answer as a number.

Numeric entry — write the number.

Q23 · NUM-DPV-045 · Select all that apply · 105s

The digits of the three-digit positive integer N are three consecutive integers in some order; for example, 213 and 654 are such integers. Which of the following must be true of every such N? Select all that apply.

(A) N is divisible by 3
(B) N is divisible by 9
(C) At least one digit of N is even
(D) No digit of N is 0
(E) N is divisible by the sum of its digits

Q24 · NUM-DPV-046 · MCQ · 100s

How many three-digit positive integers have digits that sum to 4?

(A) 3
(B) 4
(C) 10
(D) 13
(E) 15

Q25 · NUM-DPV-047 · QC · 110s

a is the tens digit and b is the units digit of a two-digit positive integer, so the integer is 10a + b.

Column A: The greatest possible value of (10a + b)/(a + b) Column B: 10

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Level 4 · Extreme — 5 questions · ~11 min

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

Q26 · NUM-DPV-038 · Numeric Entry · 130s

How many four-digit positive integers have all four digits non-zero and have the sum of their first two digits equal to the sum of their last two digits? Enter your answer as a number.

Numeric entry — write the number.

Q27 · NUM-DPV-040 · Select all that apply · 130s

A five-digit palindrome reads the same forwards and backwards, so its digits are d1, d2, d3, d2, d1 in order. For a particular such palindrome every digit is non-zero, 2(d1) + 2(d2) + d3 = 19, and d3 < d2 < d1. Which of the following are possible values of d1? Select all that apply.

(A) 5
(B) 6
(C) 7
(D) 8
(E) 9

Q28 · NUM-DPV-048 · MCQ · 120s

What is the least positive integer whose digits are all different and whose digits sum to 30?

(A) 3,999
(B) 5,889
(C) 6,789
(D) 6,798
(E) 9,876

Q29 · NUM-DPV-049 · QC · 130s

N is a three-digit positive integer whose units digit is not 0, and M is the three-digit integer formed by writing the digits of N in reverse order.

Column A: The number of such integers N for which N + M is a perfect square Column B: 8

Q30 · NUM-DPV-050 · QC · 130s

For a three-digit positive integer N, let S(N) denote the sum of the digits of N.

Column A: The number of three-digit integers N for which N - S(N) = 369 Column B: The number of three-digit integers N for which N - S(N) = 368

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