🏠 Homepage 📘 Download PDF Books 📕 Premium PDF Books

Divisibility Rules Quiz 3 (10 MCQs)

This set of multiple-choice questions evaluates understanding of divisibility rules, including divisibility by 2, 3, 4, 6, and 9, as well as arithmetic sequences. It tests skills in numerical computation, sum of digits, and identifying factors and multiples. Concepts such as even numbers, multiples of 5 and 10, and the last two digits of numbers are also covered.

Quiz Instructions

Select an option to see the correct answer instantly.

1. Which of these is divisible by 6?

Correct Answer: Correct answer is: (A) 318.

Exam Relevance: SAT, GRE, GMAT, ACT

Difficulty: Easy

Concept notes: A number is divisible by 6 if it is divisible by both 2 and 3.

Common Mistakes: A common mistake is to only check for divisibility by 2 or 3, but not both.

Explanations: To determine if 318 is divisible by 6, we need to check if it is divisible by both 2 and 3. 1. Check divisibility by 2: A number is divisible by 2 if its last digit is even. The last digit of 318 is 8, which is even. Therefore, 318 is divisible by 2. 2. Check divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3. The sum of the digits of 318 is 3 + 1 + 8 = 12, which is divisible by 3. Therefore, 318 is divisible by 3. Since 318 is divisible by both 2 and 3, it is divisible by 6.

Option Analysis:

Option A: 318 is divisible by both 2 and 3, so it is divisible by 6.
Option B: 215 is not divisible by 2 (last digit is 5, which is odd), so it is not divisible by 6.
Option C: 7000 is divisible by 2 (last digit is 0, which is even), but the sum of its digits (7 + 0 + 0 + 0 = 7) is not divisible by 3, so it is not divisible by 6.
Option D: 4765 is not divisible by 2 (last digit is 5, which is odd), so it is not divisible by 6.

Mnemonic: Divisible by 6: Even and Sum 3

2. The number 252 is divisible by 6 because:

Correct Answer: Correct answer is: (C) It's even and the sum of the digits is divisible by 3.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Moderate

Concept notes: A number is divisible by 6 if it is divisible by both 2 and 3. A number is divisible by 2 if it is even, and a number is divisible by 3 if the sum of its digits is divisible by 3.

Common Mistakes: A common mistake is to think that a number is divisible by 6 if the sum of its digits is divisible by 6, which is incorrect.

Explanations: The number 252 is even, which means it is divisible by 2. The sum of the digits of 252 is 2 + 5 + 2 = 9, which is divisible by 3. Therefore, 252 is divisible by both 2 and 3, and hence by 6.

Option Analysis:

Option A: The digits add up to 6. This is incorrect because the sum of the digits of 252 is 9, not 6.
Option B: The sum of the digits is divisible by 6. This is incorrect because the sum of the digits of 252 is 9, which is not divisible by 6.
Option C: It's even and the sum of the digits is divisible by 3. This is correct because 252 is even and the sum of its digits (9) is divisible by 3.
Option D: The sum of the digits is even. This is incorrect because the sum of the digits of 252 is 9, which is not even.

Mnemonic: Divisible by 6: Even and 3's sum.

3. Find the number of all three digit natural numbers which are divisible by 9

Correct Answer: Correct answer is: (C) 100.

Exam Relevance: SAT, GRE, GMAT, Competitive Math Exams

Difficulty: Moderate

Concept notes: To find the number of three-digit natural numbers divisible by 9, we need to identify the range of three-digit numbers and then determine how many of these numbers are divisible by 9.

Common Mistakes: A common mistake is to overlook the range of three-digit numbers or to incorrectly count the numbers divisible by 9.

Explanations: The smallest three-digit number is 100, and the largest is 999. We need to find the smallest and largest three-digit numbers divisible by 9. 1. The smallest three-digit number divisible by 9 is 108 (since 100 ÷ 9 = 11.11, and the next whole number multiple of 9 is 12 × 9 = 108). 2. The largest three-digit number divisible by 9 is 999 (since 999 ÷ 9 = 111). To find the total number of three-digit numbers divisible by 9, we use the formula for the number of terms in an arithmetic sequence: \[ n = \frac{\text{Last term} - \text{First term}}{\text{Common difference}} + 1 \] Here, the first term \( a = 108 \), the last term \( l = 999 \), and the common difference \( d = 9 \). \[ n = \frac{999 - 108}{9} + 1 = \frac{891}{9} + 1 = 99 + 1 = 100 \] Thus, there are 100 three-digit natural numbers divisible by 9.

Option Analysis:

Option A: 50 is incorrect because it underestimates the number of three-digit numbers divisible by 9.
Option B: 96 is incorrect because it does not account for the full range of three-digit numbers divisible by 9.
Option C: 100 is correct as calculated above.
Option D: 105 is incorrect because it overestimates the number of three-digit numbers divisible by 9.

Mnemonic: Divide, Subtract, Divide, Add One (DSDA)

4. A farmer has an equal number of cows in 3 different fields of his farm. Which choice could be the total number of cows?

Correct Answer: Correct answer is: (C) 207.

Exam Relevance: SAT, GRE, GMAT, ACT

Difficulty: Moderate

Concept notes: The problem involves determining if a number is divisible by 3. A number is divisible by 3 if the sum of its digits is divisible by 3.

Common Mistakes: A common mistake is to assume that a number is divisible by 3 based on its last digit, which is incorrect. The divisibility rule for 3 involves the sum of the digits.

Explanations: To determine if 207 is divisible by 3, we sum its digits: 2 + 0 + 7 = 9. Since 9 is divisible by 3, 207 is also divisible by 3. Therefore, 207 could be the total number of cows.

Option Analysis:

Option A: 112. The sum of the digits is 1 + 1 + 2 = 4, which is not divisible by 3.
Option B: 178. The sum of the digits is 1 + 7 + 8 = 16, which is not divisible by 3.
Option C: 207. The sum of the digits is 2 + 0 + 7 = 9, which is divisible by 3.
Option D: 266. The sum of the digits is 2 + 6 + 6 = 14, which is not divisible by 3.

Mnemonic: Sum and check, divisible by 3.

5. Which is not a factor of 16?

Correct Answer: Correct answer is: (B) 3.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Easy

Concept notes: A factor of a number is an integer that divides the number without leaving a remainder.

Common Mistakes: A common mistake is to assume that all small numbers are factors of a given number without checking for divisibility.

Explanations: To determine if a number is a factor of 16, we need to check if 16 can be divided by the number without leaving a remainder. 16 is divisible by 1, 2, 4, 8, and 16. Since 16 divided by 3 leaves a remainder, 3 is not a factor of 16.

Option Analysis:

Option A: 1 is a factor of 16 because 16 divided by 1 equals 16 with no remainder.
Option B: 3 is not a factor of 16 because 16 divided by 3 leaves a remainder of 1.
Option C: 4 is a factor of 16 because 16 divided by 4 equals 4 with no remainder.
Option D: 8 is a factor of 16 because 16 divided by 8 equals 2 with no remainder.

6. Which of the following numbers is NOT divisible by 4?

Correct Answer: Correct answer is: (B) 1566.

Exam Relevance: SAT, GRE, GMAT, ACT

Difficulty: Moderate

Concept notes: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.

Common Mistakes: A common mistake is to check only the last digit instead of the last two digits for divisibility by 4.

Explanations: To determine if a number is divisible by 4, we need to check if the number formed by its last two digits is divisible by 4. Let's apply this rule to each option: - For 1000, the last two digits are 00, and 00 is divisible by 4. - For 1566, the last two digits are 66, and 66 is not divisible by 4. - For 5740, the last two digits are 40, and 40 is divisible by 4. - For 2024, the last two digits are 24, and 24 is divisible by 4. Since 66 is not divisible by 4, 1566 is not divisible by 4.

Option Analysis:

Option A: 1000 is divisible by 4 because 00 is divisible by 4.
Option B: 1566 is not divisible by 4 because 66 is not divisible by 4.
Option C: 5740 is divisible by 4 because 40 is divisible by 4.
Option D: 2024 is divisible by 4 because 24 is divisible by 4.

Mnemonic: Last two digits, check for 4.

7. Divided by 3

Correct Answer: Correct answer is: (B) 54.

Exam Relevance: SAT, GRE, GMAT, ACT

Difficulty: Easy

Concept notes: A number is divisible by 3 if the sum of its digits is divisible by 3.

Common Mistakes: A common mistake is to assume that a number is divisible by 3 based on its last digit, which is incorrect. The divisibility rule for 3 involves the sum of all digits.

Explanations: To determine if a number is divisible by 3, we sum its digits and check if the sum is divisible by 3. For 54, the sum of the digits is 5 + 4 = 9. Since 9 is divisible by 3, 54 is divisible by 3.

Option Analysis:

Option A: 79. The sum of the digits of 79 is 7 + 9 = 16, which is not divisible by 3. Therefore, 79 is not divisible by 3.
Option B: 54. The sum of the digits of 54 is 5 + 4 = 9, which is divisible by 3. Therefore, 54 is divisible by 3.

Mnemonic: Sum and check, divisible by 3.

8. If you add a multiple of 10 to amultiple of 5 the answer is always a multipleof 5.

Correct Answer: Correct answer is: (A) I agree.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Easy

Concept notes: A multiple of 10 is always a multiple of 5 because 10 itself is a multiple of 5. Adding a multiple of 10 to a multiple of 5 will always result in a number that is a multiple of 5.

Common Mistakes: A common misunderstanding might be that adding a multiple of 10 to a multiple of 5 could result in a number that is not a multiple of 5, but this is incorrect.

Explanations: A multiple of 10 can be written as \(10k\) where \(k\) is an integer. A multiple of 5 can be written as \(5m\) where \(m\) is an integer. Adding these together gives \(10k + 5m = 5(2k + m)\). Since \(2k + m\) is an integer, the sum \(5(2k + m)\) is a multiple of 5.

Option Analysis:

Option A: This option is correct because the sum of a multiple of 10 and a multiple of 5 is always a multiple of 5.
Option B: This option is incorrect because the sum of a multiple of 10 and a multiple of 5 is always a multiple of 5.

9. Which of the following number divisible by 3

Correct Answer: Correct answer is: (C) 234.

Exam Relevance: SAT, GRE, GMAT, ACT

Difficulty: Easy

Concept notes: A number is divisible by 3 if the sum of its digits is divisible by 3.

Common Mistakes: A common mistake is to confuse the divisibility rule for 3 with the rule for 9, which requires the sum of the digits to be divisible by 9.

Explanations: To determine if 234 is divisible by 3, we sum its digits: 2 + 3 + 4 = 9. Since 9 is divisible by 3, 234 is also divisible by 3.

Option Analysis:

Option A: 332: Sum of digits = 3 + 3 + 2 = 8, which is not divisible by 3.
Option B: 233: Sum of digits = 2 + 3 + 3 = 8, which is not divisible by 3.
Option C: 234: Sum of digits = 2 + 3 + 4 = 9, which is divisible by 3.
Option D: 101: Sum of digits = 1 + 0 + 1 = 2, which is not divisible by 3.

10. At DisneyWorld, over a 122 days period, 50,874 passengers rode the Tea Cups Ride. I want to find out how many passenger rode the Tea Cups per day. In this scenario, the passengers represent which of the following?

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

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Moderate

Concept notes: In division, the dividend is the number that is being divided. The divisor is the number by which the dividend is divided, and the quotient is the result of the division.

Common Mistakes: A common mistake is confusing the dividend with the divisor or the quotient. The dividend is the total number of items to be divided, which in this case is the total number of passengers.

Explanations: In the given scenario, the total number of passengers (50,874) is the dividend because it is the number being divided by the number of days (122) to find the number of passengers per day. The number of days (122) is the divisor, and the result of the division will be the quotient, which represents the number of passengers per day.

Option Analysis:

Option A: Correct. The total number of passengers (50,874) is the dividend.
Option B: Incorrect. The number of days (122) is the divisor, not the passengers.
Option C: Incorrect. The quotient is the result of the division, not the passengers.
Option D: Incorrect. The passengers are the dividend.

Mnemonic: Dividend: "D" for "Divided."

Frequently Asked Questions

What are divisibility rules?

Divisibility rules are shortcuts to determine if one number can be evenly divided by another without performing the actual division.

How can I use divisibility rules to check if a number is divisible by 6?

A number is divisible by 6 if it is divisible by both 2 and 3. Check if the number is even (divisible by 2) and if the sum of its digits is divisible by 3.

What is the significance of the last two digits in divisibility rules?

The last two digits of a number are used to determine if it is divisible by 4 or 25. If the last two digits form a number divisible by 4, the entire number is divisible by 4.

How do divisibility rules apply to arithmetic sequences?

In an arithmetic sequence, divisibility rules can help identify patterns in the sequence, such as whether terms are divisible by a specific number, based on the common difference and the starting term.

What is the relationship between factors and divisibility?

Factors are numbers that divide another number without leaving a remainder. Divisibility rules help identify factors of a number by checking if the number can be evenly divided by potential factors.

🏠 Back to Homepage 📘 Download PDF Books 📕 Premium PDF Books