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Magazine Chapter 5: Problem 8
Let \(m\) and \(n\) be integers such that \(m \operatorname{div} 8=s, \quad m \bmod 8=3, \quad n \operatorname{div} 8=t, \quad n \bmod 8=6\) Determine (a) \((m+2) \operatorname{div} 8\) (b) \((m+2) \bmod 8\) (c) \((3 m n) \operatorname{div} 8\) (d) \((3 m n) \bmod 8\) (e) \((5 m+2 n) \operatorname{div} 8\) (f) \((5 m+2 n) \bmod 8\) (g) \((3 m-2 n) \operatorname{div} 8\) (h) \((3 m-2 n) \bmod 8\)
Short Answer
Expert verified
a) s, b) 5, c) Simplified complex form, d) 6, e) Approximate form, f) 3, g) 3s-2t, h) 5.
Step by step solution
01
Understanding division and modulus
We know that for any integer, the operations `div` and `mod` can be represented as follows: - If an integer \( x \) applies the operation \( x \div y = q \) and \( x \bmod y = r \), then \( x = q imes y + r \). Hence, \( m = 8s + 3 \) and \( n = 8t + 6 \). This is essential for solving the steps ahead.
02
Calculate \((m+2)\div 8\)
Given that \( m = 8s + 3 \), calculate \( m+2 = 8s + 3 + 2 = 8s + 5 \). Perform \( (8s + 5) \div 8 \):Since this is simply dividing an expression by 8 where \( s \) is already known as a quotient, \( (8s + 5) \div 8 = s \) (because 5 divided by 8 results in 0).
03
Calculate \((m+2)\bmod 8\)
Now, calculate \( m + 2 \bmod 8 \). Since \( m + 2 = 8s + 5 \), dividing by 8 gives a remainder of 5: \((8s + 5) \bmod 8 = 5\).
04
Calculate \((3mn)\div 8\)
First, calculate \( 3mn = 3(8s+3)(8t+6) \). Expand this and you will notice term simplification due to the modulus operation:\[ 3mn = 192st + 144s + 72t + 54 \]. Only terms divisible by 8 contribute to the quotient, hence simplify:\( (3mn) \div 8 = 24st + 18s + 9t + 6 \).
05
Calculate \((3mn)\bmod 8\)
From \( 3mn = 192st + 144s + 72t + 54 \), reduce each coefficient by modulus 8:- \( 54 \bmod 8 = 6 \)Thus, \((3mn) \bmod 8 = 6\).
06
Calculate \((5m+2n)\div 8\)
Calculate the expression: \( 5m + 2n = 5(8s+3) + 2(8t+6) = 40s + 15 + 16t + 12 \) simplify:\( = 40s + 16t + 27 \). So, it can further be calculated:\( (40s + 16t + 27) \div 8 \approx 5s + 2t + 3 \).
07
Calculate \((5m+2n)\bmod 8\)
From \( (5m+2n) = 40s + 16t + 27 \), find the modulus:- \( 27 \bmod 8 = 3 \)Thus, \((5m+2n) \bmod 8 = 3\).
08
Calculate \((3m-2n)\div 8\)
Calculate the expression:\( 3m - 2n = 3(8s+3) - 2(8t+6) = 24s + 9 - 16t - 12 \), simplify:\( = 24s - 16t - 3 \). So: \( (24s - 16t - 3) \div 8 = 3s - 2t \).
09
Calculate \((3m-2n)\bmod 8\)
From \( 3m-2n = 24s - 16t - 3 \), find the modulus:- \( -3 \bmod 8 = 5 \) (because the remainder is always non-negative and adds 8 in computation)Thus, \((3m-2n) \bmod 8 = 5\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integer Division
Integer division is a fundamental concept in mathematics, often used in computer science and related fields. When you perform integer division, you're dividing one integer by another and discarding the remainder. This means you only care about how many whole times the divisor fits into the dividend. For example, if you divide 17 by 4, the result of integer division is 4, because 4 goes into 17 a total of 4 whole times, with a remainder leftover.
- The operation is often denoted by \( x \div y = q \), where \( q \) is the quotient.
- It's important to remember that in many programming languages, like Python and Java, the division symbol for integer division is \( // \).
Modulus Operation
The modulus operation is used to find the remainder after division of one number by another. It's denoted by the symbol \( \bmod \). In the equation \( a \bmod n \), the result is the remainder when \( a \) is divided by \( n \). This operation is commonly used in contexts like determining whether a number divides evenly by another (used in checks like even or odd numbers).
- For instance, \( 17 \bmod 5 = 2 \) because when 17 is divided by 5, the quotient is 3 with a remainder of 2.
- It helps in cyclical operations or periodicity, such as finding angles in geometry, days of the week calculations, and more.
Remainder Theorem
The remainder theorem states that any integer \( x \) can be expressed in terms of another integer \( y \), a quotient \( q \), and a remainder \( r \), such that \( x = y \, q + r \). Here, the remainder \( r \) must be less than \( y \) and greater than or equal to 0. This fundamental principle is heavily used in modular arithmetic, helping break down more complex equations into easier pieces that are easier to work with.
- For example, when we say \( 26 = 5 \, 5 + 1 \), it means that 26 divided by 5 gives a quotient of 5 and a remainder of 1.
Step-by-Step Solution
The step-by-step solution approach breaks down complex problems into smaller, manageable parts, making it easier to solve mathematically sophisticated problems. Here, each part of the problem is tackled individually, using core concepts like integer division and the modulus operation to systematically solve for different expressions.
- This approach is productive in modular arithmetic, where equations can become intricate and prone to mistakes if tackled all at once.
- By following this method, one ensures that each step logically follows from the previous one, reducing errors and confusion.
