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Number System Quiz 26 (10 MCQs)

This set of multiple-choice questions evaluates understanding of divisibility rules, prime number identification, square root approximation, and number system concepts such as absolute value, number representation, and place value. It also tests skills in comparing negative numbers, number ordering, and basic arithmetic operations.

Quiz Instructions

Select an option to see the correct answer instantly.

1. Eighty

Correct Answer: Correct answer is: (A) 80.

Exam Relevance: SAT, GRE, GMAT, Competitive Exams

Difficulty: Very Easy

Concept notes: In the number system, the number eighty is represented as 80.

Common Mistakes: A common mistake is to confuse the order of digits, such as writing 08 instead of 80.

Explanations: The number eighty is written as 80 in the decimal number system. The digit 8 represents eighty, and the digit 0 represents zero units. Therefore, the correct representation of eighty is 80.

Option Analysis:

Option A: 80 is the correct representation of the number eighty.
Option B: 08 is not the standard representation of eighty; it is typically written as 80.
Option C: 800 represents eight hundred, not eighty.
Option D: None of the above is incorrect because 80 is a valid option.

2. What is the $\sqrt[]{64}$

Correct Answer: Correct answer is: (A) $\pm8$.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Very Easy

Concept notes: The square root of a number is a value that, when multiplied by itself, gives the original number. For 64, the square root is 8 because $8 \times 8 = 64$. Since the square root can be both positive and negative, the answer is $\pm8$.

Common Mistakes: A common mistake is to only consider the positive square root and ignore the negative square root.

Explanations: The square root of 64 is a value that, when squared, equals 64. Since $8 \times 8 = 64$ and $(-8) \times (-8) = 64$, the square root of 64 is $\pm8$.

Option Analysis:

Option A: Correct. The square root of 64 is $\pm8$.
Option B: Incorrect. The square root of 64 is not $\pm4$.
Option C: Incorrect. The square root of 64 is not $\pm16$.
Option D: Incorrect. The square root of 64 is not $\pm10$.

3. Which of the following numbers is a prime number?

Correct Answer: Correct answer is: (A) 167.

Exam Relevance: SAT, GRE, GMAT, CAT

Difficulty: Moderate

Concept notes: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Common Mistakes: A common mistake is to assume that all odd numbers are prime, which is not true. For example, 169 is not prime because it is divisible by 13.

Explanations: To determine if 167 is a prime number, we need to check if it has any divisors other than 1 and itself. We can do this by checking divisibility by all prime numbers less than or equal to the square root of 167. The square root of 167 is approximately 12.92, so we only need to check divisibility by the prime numbers 2, 3, 5, 7, and 11. - 167 is not divisible by 2 because it is odd. - The sum of the digits of 167 is 1 + 6 + 7 = 14, which is not divisible by 3, so 167 is not divisible by 3. - 167 does not end in 0 or 5, so it is not divisible by 5. - To check divisibility by 7, we perform the division: 167 ÷ 7 ≈ 23.857, which is not an integer, so 167 is not divisible by 7. - To check divisibility by 11, we perform the division: 167 ÷ 11 ≈ 15.182, which is not an integer, so 167 is not divisible by 11. Since 167 is not divisible by any of these prime numbers, it is a prime number.

Option Analysis:

Option A: 167 is a prime number as shown by the divisibility checks.
Option B: 168 is not a prime number because it is divisible by 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, and 168.
Option C: 169 is not a prime number because it is 13 × 13.
Option D: 170 is not a prime number because it is divisible by 2, 5, 10, 17, 34, 85, and 170.

Mnemonic: Prime numbers are "lonely" because they have no other friends (divisors) except 1 and themselves.

4. The predecessor of 721,896 is .....

Correct Answer: Correct answer is: (D) 721,895.

Exam Relevance: SAT, GRE, GMAT, Competitive Exams

Difficulty: Very Easy

Concept notes: The predecessor of a number is the number that comes immediately before it in the number sequence.

Common Mistakes: A common mistake is to confuse the predecessor with the successor, which is the number that comes immediately after the given number.

Explanations: The predecessor of 721,896 is the number that comes immediately before it. To find the predecessor, we subtract 1 from 721,896. Therefore, 721,896 - 1 = 721,895.

Option Analysis:

Option A: 721,899 is incorrect because it is greater than 721,896.
Option B: 721,897 is incorrect because it is greater than 721,896.
Option C: 721,877 is incorrect because it is not the immediate predecessor of 721,896.
Option D: 721,895 is correct because it is the immediate predecessor of 721,896.

5. Put the following in standard form. Five hundred thirty-four thousand, forty-five.

Correct Answer: Correct answer is: (B) 534,045.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Easy

Concept notes: The standard form of a number is its representation in a way that clearly shows the place value of each digit.

Common Mistakes: A common mistake is misplacing the digits, particularly the zeros, which can lead to incorrect standard form.

Explanations: To convert "five hundred thirty-four thousand, forty-five" to standard form, we break it down as follows: - "Five hundred thirty-four thousand" is 534,000. - "Forty-five" is 45. - Combining these, we get 534,045.

Option Analysis:

Option A: 53,445 is incorrect because it does not represent the correct place values for the given number.
Option B: 534,045 is correct as it accurately represents the number "five hundred thirty-four thousand, forty-five" in standard form.
Option C: 534,450 is incorrect because it places the digits in the wrong positions.
Option D: None of the above is incorrect because option B is the correct answer.

6. Which number is the smallest?-3, |-520|,-520, or 3

Correct Answer: Correct answer is: (C) -520.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Moderate

Concept notes: In the number system, negative numbers are smaller than positive numbers. Among negative numbers, the one with the larger absolute value is smaller.

Common Mistakes: A common mistake is to confuse the absolute value of a number with its actual value. For example, |-520| is 520, which is a positive number and thus larger than -520.

Explanations: To determine the smallest number, we need to compare the values of -3, |-520|, -520, and 3. The absolute value of -520 is 520, which is a positive number. Therefore, |-520| = 520. Among the numbers -3, 520, -520, and 3, the smallest number is -520 because it has the largest absolute value among the negative numbers.

Option Analysis:

Option A: -3 is a negative number but its absolute value is smaller than that of -520, making it larger than -520.
Option B: |-520| is 520, which is a positive number and thus larger than any negative number.
Option C: -520 is the smallest number because it has the largest absolute value among the negative numbers.
Option D: 3 is a positive number and thus larger than any negative number.

7. Yes or No?The number 8 is a perfect square?

Correct Answer: Correct answer is: B) No.

Exam Relevance: SAT, GRE, GMAT, ACT

Difficulty: Easy

Concept notes: A perfect square is a number that can be expressed as the square of an integer.

Common Mistakes: A common mistake is to confuse perfect squares with other types of numbers, such as prime numbers or even numbers.

Explanations: The number 8 is not a perfect square because there is no integer $ n $ such that $ n^2 = 8 $. The closest perfect squares are 4 ($2^2$) and 9 ($3^2$).

Option Analysis:

Option A: Incorrect because 8 is not a perfect square.
Option B: Correct because 8 cannot be expressed as the square of an integer.

8. How many tens are required to make the number 49?

Correct Answer: Correct answer is: (A) 4 tens.

Exam Relevance: SAT, GRE, GMAT, Competitive Exams

Difficulty: Easy

Concept notes: In the number system, the number 49 can be broken down into tens and units. The digit in the tens place represents the number of tens, and the digit in the units place represents the number of units.

Common Mistakes: A common mistake is to focus on the units place (9) instead of the tens place (4), leading to the incorrect answer of 9 tens.

Explanations: The number 49 consists of 4 tens and 9 units. The digit 4 in the tens place indicates that there are 4 tens in the number 49. Therefore, the number of tens required to make 49 is 4.

Option Analysis:

Option A: Correct. The number 49 has 4 tens.
Option B: Incorrect. The number 49 has 9 units, not 9 tens.

9. 5tens+8ones=

Correct Answer: Correct answer is: (A) 58.

Exam Relevance: Common Core Math, SAT, ACT

Difficulty: Easy

Concept notes: In the number system, tens and ones represent place values. The tens place is the second digit from the right, and the ones place is the first digit from the right.

Common Mistakes: A common mistake is to reverse the digits, placing the ones in the tens place and the tens in the ones place, resulting in 85 instead of 58.

Explanations: The number 5 tens is equivalent to 50, and 8 ones is equivalent to 8. Adding these together, we get 50 + 8 = 58.

Option Analysis:

Option A: 58 is the correct sum of 5 tens and 8 ones.
Option B: 85 is incorrect because it reverses the place values of tens and ones.

10. Write the expanded form:433 =

Correct Answer: Correct answer is: (A) 400 + 30 + 3.

Exam Relevance: Common Core Math, SAT, ACT, GRE

Difficulty: Easy

Concept notes: The expanded form of a number breaks it down into the sum of its place values.

Common Mistakes: A common mistake is to misinterpret the place values, such as placing the hundreds and tens digits in the wrong order.

Explanations: The number 433 can be broken down into its place values: - The digit 4 is in the hundreds place, so it represents 400. - The digit 3 is in the tens place, so it represents 30. - The digit 3 is in the ones place, so it represents 3. Thus, the expanded form of 433 is 400 + 30 + 3.

Option Analysis:

Option A: Correct. 400 + 30 + 3 is the correct expanded form of 433.
Option B: Incorrect. 300 + 30 + 4 does not represent the correct place values of 433.
Option C: Incorrect. 400 + 40 + 3 does not represent the correct place values of 433.
Option D: Incorrect. None of the above is not applicable since Option A is correct.

Frequently Asked Questions

What is the significance of place value in the number system?

Place value determines the value of a digit based on its position in a number. For example, in the number 234, the digit 4 is in the ones place, 3 is in the tens place, and 2 is in the hundreds place.

How do you determine if a number is a perfect square?

A number is a perfect square if its square root is an integer. For example, 16 is a perfect square because its square root is 4, which is an integer.

What are the rules for determining divisibility by 3?

A number is divisible by 3 if the sum of its digits is divisible by 3. For example, the number 123 is divisible by 3 because 1 + 2 + 3 = 6, and 6 is divisible by 3.

How do you represent negative numbers in the number system?

Negative numbers are represented with a minus sign (-) before the number. For example, -5 is a negative number, indicating a value five units below zero.

What is the expanded form of a number?

The expanded form of a number shows the value of each digit. For example, the expanded form of 345 is 300 + 40 + 5.

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