Check whether 61479 is divisible by 81

Is 61479 Divisible by 81? A Step-by-Step Explanation

To determine whether the number 61479 is divisible by 81, we use basic divisibility rules and arithmetic. Let’s explore the solution step by step.

What Does Divisibility Mean?

A number is divisible by another if dividing them results in a whole number with no remainder. For example:

• 27 ÷ 9 = 3 (No remainder, divisible)
• 28 ÷ 9 = 3.11 (Not a whole number, not divisible)

For 61479 and 81, we need to check if dividing them produces a whole number.

Step 1: Perform the Division

Divide 61479 by 81:

$$61479÷81=759$$

The result is a whole number (759) with no remainder, indicating that 61479 is divisible by 81.

Step 2: Verify Using Multiplication

To confirm, multiply the result (759) by 81:

$$759\times 81=61479$$

Since the product equals the original number, the calculation is accurate.

Step 3: Understanding Divisibility by 81

81 is a power of 3 (3⁴ = 81). If a number is divisible by 81, it must also be divisible by 3 four times consecutively. This is evident in 61479 because:

1. The sum of the digits of 61479 is $6+1+4+7+9=27$.
2. 27 is divisible by 3 (as $27÷3=9$).
3. Repeated divisibility by 3 confirms that 61479 passes the test for divisibility by 81.

Is 61479 Divisible by 81? A Comprehensive Analysis

Understanding whether a number like 61479 is divisible by 81 involves straightforward calculations and mathematical principles. Let’s delve deeper into the process, explaining every step for clarity and verification.

What Does Divisibility by 81 Imply?

For a number to be divisible by 81, dividing it by 81 should result in:

1. A whole number (no decimal or fraction).
2. A remainder of zero.

For 61479, we’ll test its divisibility by using direct division, multiplication verification, and mathematical properties of 81.

Step 1: Perform the Division

First, divide 61479 by 81 using long division or a calculator:

$$61479÷81=759$$

• The quotient is a whole number, 759.
• The remainder is 0, confirming that 61479 is divisible by 81.

Step 2: Verify with Multiplication

To ensure accuracy, multiply the quotient (759) back by 81:

$$759\times 81=61479$$

Since the result matches the original number, it confirms our division was correct.

Step 3: Check Divisibility by 3 and Powers of 3

81 is a power of 3, specifically 34=813^4 = 8134=81. For a number to be divisible by 81, it must also be divisible by 3 four times consecutively. We use the following steps:

(i) Sum of the Digits

The sum of the digits of 61479 is:

$$6+1+4+7+9=27$$

A number is divisible by 3 if the sum of its digits is divisible by 3. Here, $27÷3=9$, so 27 is divisible by 3.

(ii) Test for Successive Divisibility by 3

We divide the digit sum (27) by 3 repeatedly to ensure divisibility up to 343^434:

1. $27÷3=9$ (divisible)
2. $9÷3=3$ (divisible)
3. $3÷3=1$ (divisible)

Since we can divide by 3 four times, it confirms that 61479 is divisible by 34=813^4 = 8134=81.

Step 4: Understanding Practical Applications

Divisibility checks like this are essential in various mathematical and real-world scenarios:

• Factoring large numbers: Knowing a number’s divisibility by factors like 81 simplifies its breakdown into smaller components.
• Cryptography and encoding: Large numbers are often tested for divisibility in encryption algorithms.
• Numerical analysis: Divisibility helps verify patterns or properties in datasets.

Final Confirmation: Yes, 61479 is Divisible by 81

Through division, multiplication verification, and the rule of successive divisibility by 3, we confirm that 61479 is divisible by 81. The quotient is 759, and there is no remainder. This makes 61479 a perfect multiple of 81.