Divisibility Rules for 3, 6 and 9: Practice Problems

Master divisibility rules for 3, 6, and 9 with step-by-step examples and practice problems. Learn digit sum tricks and shortcuts for quick mental math.

📚What You'll Master in This Practice Session
  • Apply the digit sum rule to test divisibility by 3
  • Recognize when numbers are divisible by 6 using combined rules
  • Use the digit sum method for divisibility by 9
  • Identify divisibility patterns in large numbers quickly
  • Solve real-world problems using divisibility shortcuts
  • Build confidence with mental math divisibility tricks

Understanding Divisibility Rules for 3, 6 and 9

Complete explanation with examples

Divisibility criteria for 33, 66 and 99

Divisibility criteria for 33

A number is divisible by 33 if the sum of its digits is a multiple of 33.

Divisibility criteria for 66

A number is divisible by 66 if it is even and also a multiple of 33.

Divisibility criteria for99

A number is divisible by 99 if the sum of its digits is a multiple of 99.

Color-coded chart of divisibility rules from 2 to 10, using book icons to explain how to check if a number is divisible by each digit through simple criteria like even digits, digit sums, and ending digits.

Detailed explanation

Practice Divisibility Rules for 3, 6 and 9

Test your knowledge with 7 quizzes

Is the number below divisible by 9?

\( 987 \)

Examples with solutions for Divisibility Rules for 3, 6 and 9

Step-by-step solutions included
Exercise #1

Will a number divisible by 6 necessarily be divisible by 3?

Step-by-Step Solution

To determine whether a number divisible by 6 is necessarily divisible by 3, we need to understand the properties of divisibility for the numbers involved.

Let's analyze the problem step by step:

  • Step 1: Restate the Problem
    We need to find out if any number that is divisible by 6 is also divisible by 3.
  • Step 2: Identify Key Information and Variables
    - A number is divisible by 6 if it can be expressed as k×6 k \times 6 for any integer k k .
    - We want to check if such a number is also divisible by 3, meaning it can also be expressed as m×3 m \times 3 for some integer m m .
  • Step 3: Relevant Theorems
    - A number is divisible by 6 if it is divisible by both 2 and 3.
  • Step 4: Choose Approach
    We'll use the divisibility rules for numbers to deduce if a number divisible by 6 must be divisible by 3.
  • Step 5: Steps for Solution
    1. Given a number is divisible by 6, it is expressed as a multiple of 6: n=k×6 n = k \times 6 .
    2. Since 6 can be factored into 2×3 2 \times 3 , a number divisible by 6 is also divisible by 3.
    3. Therefore, n=k×6=k×(2×3)=(k×2)×3 n = k \times 6 = k \times (2 \times 3) = (k \times 2) \times 3 , making it divisible by 3.
  • Step 6: Assumptions
    We assume the integer k k is any integer and does not affect the general proof.
  • Step 7: Conclusion
    Every number divisible by 6 is necessarily divisible by both 2 and 3, due to the factorization properties of numbers. Thus, by the rules of divisibility, a number divisible by 6 is necessarily divisible by 3.

Therefore, the answer to the problem is Yes.

Answer:

Yes

Exercise #2

Will a number divisible by 6 necessarily be divisible by 2?

Step-by-Step Solution

In order to determine if a number divisible by 6 is also divisible by 2, we first review the divisibility rules:

  • A number is divisible by 6 if it is divisible by both 2 and 3.
  • A number is divisible by 2 if its last digit is even (one of 0, 2, 4, 6, or 8).

Consider a number n n that is divisible by 6. By definition, since 6 itself factors into 2 multiplied by 3, any number divisible by 6 must be divisible by 2 and 3. This means that any number divisible by 6 is automatically divisible by 2 because 2 is a part of its factorization.

Therefore, yes, any number divisible by 6 will necessarily be divisible by 2 as per the rule of divisibility.

Thus, the correct choice is:

  • Choice 1: Yes

This conclusion adheres strictly to divisibility rules and confirms the assertion that being divisible by 6 includes being divisible by 2.

Answer:

Yes

Video Solution
Exercise #3

Determine if the following number is divisible by 3:

132 132

Step-by-Step Solution

To determine if the number 132 132 is divisible by 3 3 , we can apply the rule for divisibility by 3 3 , which involves summing the digits of the number.

Step-by-step solution:

  • Step 1: Sum the digits of the number 132 132 .
    1+3+2=6 1 + 3 + 2 = 6
  • Step 2: Check if the sum is divisible by 3 3 .
    6÷3=2 6 \div 3 = 2 , which is an integer.

Since the sum of the digits is 6 6 and 6 6 is divisible by 3 3 , the number 132 132 is also divisible by 3 3 .

Therefore, the number 132 132 is divisible by 3 3 , and the correct choice is:

Yes

Answer:

Yes

Video Solution
Exercise #4

Determine if the following number is divisible by 3:

352 352

Step-by-Step Solution

To determine if 352 is divisible by 3, we need to follow these steps:

  • Calculate the sum of its digits.
  • Check if this sum is divisible by 3.

Let's work through the procedure:

The number 352 352 consists of the digits 3, 5, and 2.

Step 1: Calculate the sum of the digits.

The sum is 3+5+2=10 3 + 5 + 2 = 10 .

Step 2: Check if 10 is divisible by 3.

Since 10 divided by 3 gives a remainder, 10 is not divisible by 3.

Therefore, the number 352 is not divisible by 3.

The correct answer is No.

Answer:

No

Video Solution
Exercise #5

Determine if the following number is divisible by 3:

673 673

Step-by-Step Solution

To determine if 673 is divisible by 3, we must use the divisibility rule for 3, which states that a number is divisible by 3 if the sum of its digits is divisible by 3.

First, we'll calculate the sum of the digits: 6+7+36 + 7 + 3.

Calculating this, we get: 6+7+3=166 + 7 + 3 = 16.

Next, we check if 16 is divisible by 3. Dividing 16 by 3 gives a quotient of 5 and a remainder of 1.

Since 16 is not divisible by 3 (as it leaves a remainder), we conclude that 673 is not divisible by 3.

Thus, the correct answer is No.

Answer:

No

Video Solution

Frequently Asked Questions

What is the divisibility rule for 3?

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

How do you check if a number is divisible by 6?

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A number is divisible by 6 if it's divisible by both 2 and 3. This means the number must be even (ends in 0, 2, 4, 6, or 8) AND the sum of its digits must be divisible by 3.

What's the fastest way to test divisibility by 9?

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Add all the digits of the number together. If this sum is divisible by 9, then the original number is divisible by 9. Keep adding digits until you get a single digit - if it's 9, the number is divisible by 9.

Why do divisibility rules for 3 and 9 work with digit sums?

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These rules work because of how our base-10 number system relates to remainders. When you divide powers of 10 by 3 or 9, the remainders follow a pattern that makes the digit sum method mathematically valid.

Can you use divisibility rules for large numbers?

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Yes! Divisibility rules work for numbers of any size. For very large numbers, these rules are much faster than long division. Simply add all the digits together and apply the same rules.

What grade level learns divisibility rules for 3, 6, and 9?

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These divisibility rules are typically introduced in 4th-6th grade math curricula. They're foundational for understanding factors, multiples, and preparing for fraction work in middle school.

Are there tricks to remember divisibility rules?

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Memory tricks include: '3 and 9 use digit sums,' '6 needs both 2 and 3 rules,' and practicing with familiar numbers like phone numbers or addresses to build automatic recognition.

How are divisibility rules used in real life?

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Divisibility rules help with: splitting items into equal groups, calculating tips and discounts, checking if measurements divide evenly, and solving everyday math problems without a calculator.

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