90 Divided By 12 With Remainder

90 Divided by 12 with Remainder: A Deep Dive into Division

The seemingly simple question, "What is 90 divided by 12 with a remainder?", opens a door to a fascinating exploration of arithmetic, number theory, and even practical applications in everyday life. While the answer itself is readily obtainable, understanding the process and the underlying mathematical principles provides a more robust understanding of division and its broader implications. This article will delve deep into this seemingly simple problem, exploring different methods of calculation, examining the concept of remainders, and highlighting its significance in various contexts.

Understanding Division and Remainders

Division, at its core, is the process of splitting a quantity into equal parts. When we divide 90 by 12, we're asking: "How many times does 12 fit completely into 90?" The result of this division, without considering the remainder, is called the quotient. However, in many instances, the divisor (12 in this case) does not divide the dividend (90) perfectly. The amount left over after the division is complete is the remainder.

This concept is formally expressed as:

Dividend = (Divisor × Quotient) + Remainder

In our example:

90 = (12 × Quotient) + Remainder

Calculating 90 Divided by 12 with Remainder

Let's explore several ways to calculate 90 divided by 12 and find the remainder:

Method 1: Long Division

Long division is a classic method for performing division, particularly useful when dealing with larger numbers. Here's how to apply it to 90 divided by 12:

1. Set up the problem: Write 90 inside the long division symbol and 12 outside.

2. Determine how many times 12 goes into 90: 12 goes into 90 seven times (12 x 7 = 84).

3. Multiply and subtract: Multiply 12 by 7 (which equals 84), and subtract this from 90 (90 - 84 = 6).

4. Identify the remainder: The result of the subtraction, 6, is the remainder.

Therefore, 90 divided by 12 is 7 with a remainder of 6.

Method 2: Repeated Subtraction

Repeated subtraction is a more intuitive approach, particularly helpful for visualizing the division process. We repeatedly subtract the divisor (12) from the dividend (90) until we reach a number smaller than the divisor. The number of times we subtract is the quotient, and the remaining number is the remainder.

1. Start with 90: Subtract 12 repeatedly: 90 - 12 = 78, 78 - 12 = 66, 66 - 12 = 54, 54 - 12 = 42, 42 - 12 = 30, 30 - 12 = 18, 18 - 12 = 6.

2. Count the subtractions: We subtracted 12 seven times.

3. Identify the remainder: The final result, 6, is the remainder.

Again, we find that 90 divided by 12 is 7 with a remainder of 6.

Method 3: Using a Calculator

While calculators provide a quick answer, they don't necessarily illuminate the underlying process. Most calculators will directly provide the quotient (7) if you input 90 ÷ 12. To find the remainder, you'd need to perform a separate calculation: multiply the quotient by the divisor (7 x 12 = 84) and subtract the result from the dividend (90 - 84 = 6).

The Significance of Remainders

Remainders aren't just leftover numbers; they hold significant meaning in various mathematical and practical contexts:

• Modular Arithmetic: Remainders are fundamental in modular arithmetic, a system where numbers "wrap around" after reaching a certain value (the modulus). This system is crucial in cryptography, computer science (hashing algorithms), and scheduling problems. In our example, 90 modulo 12 (written as 90 mod 12) equals 6.

• Even and Odd Numbers: The remainder when a number is divided by 2 determines whether it's even (remainder 0) or odd (remainder 1).

• Divisibility Rules: Remainders are used to test for divisibility by certain numbers. For example, a number is divisible by 3 if the sum of its digits is divisible by 3. The remainder when divided by 3 indicates whether the number is a multiple of 3.

• Real-world Applications: Imagine you have 90 apples and want to pack them into boxes of 12. The quotient (7) tells you how many boxes you can completely fill, and the remainder (6) indicates how many apples are left over.

Expanding the Concept: Factors, Multiples, and Prime Numbers

Understanding the relationship between 90 and 12 extends beyond simple division. Let's explore some related concepts:

• Factors: The factors of a number are the whole numbers that divide it evenly (with a remainder of 0). The factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90. The factors of 12 are 1, 2, 3, 4, 6, and 12. The greatest common factor (GCF) of 90 and 12 is 6.

• Multiples: A multiple of a number is the product of that number and any whole number. Multiples of 12 include 12, 24, 36, 48, 60, 72, 84, 96, etc. 90 is not a multiple of 12.

• Prime Factorization: Prime factorization is the process of expressing a number as a product of its prime factors (numbers divisible only by 1 and themselves). The prime factorization of 90 is 2 x 3² x 5. The prime factorization of 12 is 2² x 3. This reveals the common factors (2 and 3) that explain why 6 is the GCF.

• Prime Numbers: Understanding prime numbers is crucial in number theory and cryptography. Prime numbers are only divisible by 1 and themselves. Neither 90 nor 12 are prime numbers.

Applying Remainders in Programming

The concept of remainders finds extensive use in programming. The modulo operator (%) calculates the remainder of a division. This is commonly used for:

• Even/Odd Checks: Determining if a number is even or odd.
• Cyclic Processes: Creating repeating patterns or sequences.
• Hashing: Generating unique identifiers for data elements.
• Data Validation: Verifying input data meets specific criteria.

For example, in Python:

remainder = 90 % 12  # remainder will be 6

Conclusion: Beyond the Basics

The seemingly trivial problem of 90 divided by 12 with a remainder offers a gateway to a rich understanding of fundamental mathematical concepts. From the mechanics of long division to the theoretical underpinnings of modular arithmetic and prime factorization, this simple problem illustrates the interconnectedness of mathematical ideas and their practical applications in diverse fields. The ability to grasp the concept of remainders and apply it effectively is a valuable skill with wide-ranging implications, extending far beyond the classroom into the realms of computer science, cryptography, and even everyday problem-solving. The journey from a simple calculation to a deep understanding of number theory showcases the beauty and power of mathematics.