15 Aug, 2024

· Mathematics

What divided by what gives you 12 explained

Short Answer

Some answer Some answer Some answer

Long Explanation

Explanation

Division Basics

Division is the operation of distributing a number into specified parts. For the equation:

\frac{a}{b} = 12

We aim to identify pairs of $(a, b)$ that satisfy this equation.

Key Pairs

Some common pairs include:

$a = 24, b = 2$
$a = 36, b = 3$
$a = 48, b = 4$
$a = 60, b = 5$
$a = 144, b = 12$

General Formula

We can generalize this with:

$a = 12b$

For any real number $b \neq 0$ , $a$ is simply 12 times that number.

Multi-line Display

For more complex scenarios, the division pairs can be represented as:

\begin{aligned} &\frac{24}{2} = 12 \\[10pt] &\frac{36}{3} = 12 \\[10pt] &\frac{48}{4} = 12 \\[10pt] &\frac{60}{5} = 12 \\[10pt] &\frac{144}{12} = 12 \\ \end{aligned}

Conclusion

To achieve a quotient of 12, the numerator must always be twelve times the denominator. This relationship underpins the result, solidifying our understanding of division leading to 12.

Verified By

Emily Rosen

Mathematics Content Writer at Math AI

Emily Rosen is a recent graduate with a Master's in Mathematics from the University of Otago. She has been tutoring math students and working as a part-time contract writer for the past three years. She is passionate about helping students overcome their fear of math through easily digestible content.

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Concept

Division Basics

Introduction to Division Basics

In mathematics, division is one of the four fundamental arithmetic operations. The process involves splitting a number into equal parts. Here, we'll cover the essentials you need to understand division.

Key Terms in Division

Dividend: The number you want to divide.
Divisor: The number by which you want to divide the dividend.
Quotient: The result of the division.
Remainder: The part of the dividend that is left over, if it's not evenly divisible by the divisor.

Division Symbol

Division is often represented by the symbol $\div$ or a forward slash $/$ .

Basic Division Formula

The general form of a division operation is:

\text{Dividend} \div \text{Divisor} = \text{Quotient} + \text{Remainder}

Understanding with an Example

For instance, consider $15 \div 4$ :

Dividend: 15
Divisor: 4

Applying the division:

15 \div 4 = 3 \text{ R } 3

Here, 3 is the quotient and the remainder is also 3 because $4 \times 3 + 3 = 15$ .

Long Division Method

To solve divisions manually, especially for larger numbers, you can use the long division method. Here's a step-by-step guide:

Divide the dividend by the divisor.
Multiply the divisor by the current quotient.
Subtract the result from the current dividend.
Bring down the next digit of the dividend (if any) and repeat the process.

Example of Long Division

Let’s perform $123 \div 4$ using long division:

Divide 12 by 4: $12 \div 4 = 3$ The quotient is 3, with no remainder yet.
Multiply 3 by 4: $3 \times 4 = 12$
Subtract 12 from 12: $12 - 12 = 0$
Bring down the next digit, 3: $\downarrow 3$
Divide 3 by 4: $3 \div 4 = 0 \text{ R } 3$ The quotient here is 0, and the remainder is 3.

So, overall:

123 \div 4 = 30 \text{ R } 3

Important Points

Division by zero is undefined. That is, you cannot divide any number by zero.
Dividing zero by any non-zero number will always yield zero.
Division is the inverse operation of multiplication.

Conclusion

Understanding the basics of division is crucial for working through more complex arithmetic and algebraic problems. The key is to grasp the relationship between the dividend, divisor, quotient, and remainder, and to practice with examples to build proficiency.

Concept

Key Pairs

Explanation of Key Pairs

In cryptography, key pairs are essential components used for secure communication and data protection. A key pair consists of two mathematically related keys: a public key and a private key. These keys are utilized in asymmetric encryption, which allows for a more secure and flexible approach compared to symmetric encryption.

Components of Key Pairs

Public Key: This key is shared openly and can be distributed widely. It is used to encrypt data or verify a digital signature.
Private Key: This key is kept secret and only known to the owner. It is used to decrypt data or create a digital signature.

Asymmetric Encryption

Asymmetric encryption relies on the mathematical relationship between the public and private keys. Here is how it works:

Encryption: When someone wants to send you a secure message, they use your public key to encrypt the data. Despite having the public key, the encrypted data can only be decrypted by your private key.

\text{Encryption:}\ E_{\text{public}}(M) = C

Decryption: Once you receive the encrypted message (ciphertext $C$ ), you will use your private key to decrypt it back into the original message $M$ .

\text{Decryption:}\ D_{\text{private}}(C) = M

Digital Signatures

Key pairs are also used in digital signatures, which ensures the authenticity and integrity of a message or document.

Signing: You can use your private key to create a digital signature for a message or document, ensuring that it comes from you.

\text{Signature:}\ S_{\text{private}}(M) = \text{Signature}

Verification: Anyone with your public key can verify the signature, confirming that it was indeed signed by you and that the contents have not been altered.

\text{Verification:}\ V_{\text{public}}(\text{Signature}, M) = \text{Valid/Invalid}

Importance of Key Pairs

Key pairs are crucial for various applications, including:

Secure Communication: Ensures data is transmitted securely over potentially insecure channels.
Authentication: Confirms the identity of users and devices.
Data Integrity: Makes sure the data has not been tampered with.

By employing key pairs, modern cryptographic systems can provide a high level of security and trust, which is indispensable in today's digital world.