Divisibility rule of 4
Divisibility rule of 4
The divisibility rule for 4 provides a practical method to quickly determine if a number is divisible by 4, simply by checking if the number formed by its last two digits is divisible by 4. This rule is pivotal in various branches of mathematics, including Algebra, where it aids in factorization and simplification tasks. It also plays a role in operations such as addition, subtraction, multiplication, and division, enhancing efficiency in handling both rational and irrational numbers as well as integers. Understanding and applying this rule is essential for mastering foundational arithmetic operations and advancing in mathematical studies.
Download Proof of Divisibility Rule of 4 in PDF
What is the Divisibility Rule of 4?
Proof of Divisibility Rule of 4
Download Proof of Divisibility Rule of 4 in PDF
Step 1: Consider the number 4312.
Step 2: Let’s expand this number as shown below:
Step 3: 4312=4×1000+3×100+1×10+2×1
Step 4: =4×(999+1)+3×(99+1)+1×(9+1)+2×1
=(4×999+3×99+1×9)+(4×1+3×1+1×1+2×1)
Step 5: =(4×999+3×99+1×9)+(4+3+1+2)
Step 6: Recognize that numbers 9, 99, 999, etc., are divisible by 4, making their multiples also divisible by 4.
The terms 4×999, 3×99, and 1×9 are all divisible by 4, so we focus on the remaining term.
Step 7: Evaluate the divisibility of the sum of the last two digits:
- The digits are 1 and 2, forming the number 12.
- Check if 12 is divisible by 4: 12÷4 = 3, which is an integer.
Step 8: Conclusion from the example:
- Since the number formed by the last two digits, 12, is divisible by 4, the entire number 4312 is divisible by 4.
Divisibility Rule of 4 and 6
Divisibility Rule of 4
The divisibility rule for 4 states that a number is divisible by 4 if the number formed by its last two digits is divisible by 4. This rule is quite straightforward and can be quickly applied:
- Example: Consider the number 3128. The last two digits form the number 28. Since 28 is divisible by 4 (28 ÷ 4 = 7), the entire number 3128 is also divisible by 4.
Divisibility Rule of 6
The divisibility rule for 6 combines the rules for 2 and 3. A number is divisible by 6 if it is divisible by both 2 and 3. This means:
- The number must be even (i.e., it ends in 0, 2, 4, 6, or 8), which satisfies the rule for 2.
- The sum of its digits must be divisible by 3.
- Example: Consider the number 234. First, it is even, satisfying the rule for 2. Second, adding its digits gives 2 + 3 + 4 = 9, which is divisible by 3. Thus, 234 is divisible by 6 because it meets the criteria for both divisibility by 2 and by 3.
Divisibility Test of 4 and 8
The divisibility test for 4 is based on the property of the last two digits of a number. A number is divisible by 4 if the last two digits form a number that is divisible by 4. This test is particularly useful because it eliminates the need to divide the entire number, making the process much quicker.
- Example: Consider the number 3248. The last two digits are 48, which is divisible by 4 since 48÷4 = 12. Hence, 3248 is divisible by 4.
Divisibility Rule of 4 Examples
Number: 1328
- Last Two Digits: 28
- Check: Since 28 is divisible by 4 (28 ÷ 4 = 7), 1328 is divisible by 4.
Number: 3456
- Last Two Digits: 56
- Check: 56 is divisible by 4 (56 ÷ 4 = 14), therefore, 3456 is divisible by 4.
Number: 1124
- Last Two Digits: 24
- Check: 24 is divisible by 4 (24 ÷ 4 = 6), hence 1124 is divisible by 4.
Number: 9072
- Last Two Digits: 72
- Check: 72 is divisible by 4 (72 ÷ 4 = 18), so 9072 is divisible by 4.
Number: 2590
- Last Two Digits: 90
- Check: 90 is not divisible by 4 (90 ÷ 4 = 22.5), so 2590 is not divisible by 4.
FAQs
How can I quickly determine if a number like 1024 is divisible by 4?
Look at the last two digits; 24 is divisible by 4, so 1024 is also divisible by 4.
Can this rule be applied to any number regardless of length?
Yes, the divisibility rule of 4 can be applied to any number, no matter how many digits it has, as long as you can check the last two digits.
Is the number 123456 divisible by 4?
The last two digits are 56, which is divisible by 4, so yes, 123456 is divisible by 4.
What if the last two digits of a number form 00; is the number divisible by 4?
Yes, any number whose last two digits are 0 is divisible by 4, as 0 represents zero, which is divisible by any non-zero integer.
Why does the divisibility rule for 4 only consider the last two digits?
In base 10, each power of 10 increases by one zero, making any number divisible by 100 automatically divisible by 4. Thus, only the last two digits affect divisibility by 4.
Does this rule work with negative numbers?
Yes, the divisibility rule for 4 applies to negative numbers as well. Just consider the absolute value of the last two digits.
Can you provide an example where a number is not divisible by 4?
Consider the number 718. The last two digits are 18, which is not divisible by 4, so 718 is not divisible by 4.
What happens if the last two digits are 12? Is the number divisible by 4?
Yes, if the last two digits of a number are 12, the number is divisible by 4 because 12 ÷ 4 = 3 with no remainder.
How does this rule help in mathematical computations or real-world applications?
This rule simplifies the task of finding factors and multiples, aids in reducing fractions, and can be useful in practical scenarios like dividing items into groups or calculating perimeters in units divisible by four.
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