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Chapter 4: Problem 7

a) Find three positive integers \(a, b, c\) such that \(31 \mid(5 a+7 b+11 c)\). b) If \(a, b, c \in \mathbf{Z}\) and \(31 \mid(5 a+7 b+11 c)\), prove that (i) \(31 \mid(21 a+17 b+9 c)\) and (ii) \(31 \mid(6 a+27 b+7 c)\).

Short Answer

Expert verified
For part (a), the solution is a = 31, b = 31, c = 31. For part (b), the two given conditions hold true if 31 divides the expression 5a+7b+11c.

Step by step solution

01

Solution to Part (a)

The aim is to find three positive integers \(a, b, c\) such that \(5a+7b+11c\) is divisible by \(31\). For convenience, we can set \(a = 31\), \(b = 31\) and \(c = 31\). This will yield \(5(31)+7(31)+11(31)\), which is equivalent to \(31(23)\). From here, it is clear that \(31\) can divide the sum \(5a+7b+11c\). Thus, the solution to this part is \(a = 31, b = 31, c = 31\).
02

Proof for Equation (i)

In order to prove the claim that if \(31 \mid(5 a+7 b+11 c)\), then \(31 \mid(21 a+17 b+9 c)\), observe that by rearranging the terms, we get: \[21a+17b+9c = 4a+10b-2c +31a+7b+11c \] From the original statement, we know that \(31 \mid(5a+7b+11c)\), and also it is clear that \(31 \mid(31a+7b+11c)\). Hence, if \(31 \mid(5a+7b+11c)\), then also \(31 \mid(21 a+17 b+9 c)\).
03

Proof for Equation (ii)

For this step, we will show that if \(31 \mid(5 a+7 b+11 c)\), then \(31 \mid(6 a+27 b+7 c)\). By rearranging again in a similar way as before we get:\[6a+27b+7c = a+20b-4c+5a+7b+11c \] The same reasoning is applied as in step 2, which leads to the conclusion that \(31 \mid(6 a+27 b+7 c)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Divisibility
Divisibility is a fundamental concept in number theory. It refers to whether one integer can be divided by another without leaving a remainder. In our exercise, we are asked to find integers \( a, b, c \) such that a linear combination of these integers is divisible by \( 31 \). This establishes that the expression \( 5a + 7b + 11c \) is perfectly divisible by \( 31 \). To determine this, a common strategy is to check if substituting specific values satisfies the divisibility rule. For example, setting \( a = 31, b = 31, \) and \( c = 31 \) simplifies our equation and ensures divisibility by \( 31 \). This is because each term is a multiple of \( 31 \), and summing them maintains that divisibility.
  • A number \( n \) is divisible by another number \( m \) if there exists an integer \( k \) such that \( n = km \).
  • Ensuring that all terms in an expression meet this criterion guarantees the whole expression's divisibility.
Integer Solutions
Finding integer solutions means identifying values that are whole numbers or integers, satisfying a given condition. In math problems, particularly those involving divisibility, we often need to choose integers that satisfy certain divisibility criteria. For the exercise, the integers \( a = 31, b = 31, \) and \( c = 31 \) were chosen because substituting them into the expression \( 5a + 7b + 11c \) results in a number directly divisible by \( 31 \). This approach simplifies the work by leveraging known multiples of the divisor and crucially, these numbers were chosen for their simplicity and direct multiplication by \( 31 \).
  • Integer solutions refer to resolving equations or conditions where solutions must be whole numbers.
  • Choosing smart, sometimes repetitive values ensures simplicity, especially when demonstrating divisibility.
Mathematical Proofs
Mathematical proofs are systematic, logical arguments used to verify the truth of mathematical statements. They help demonstrate that certain conditions lead to specific outcomes, as shown in parts (i) and (ii) of the exercise. When we say \( 31 \mid(5 a+7 b+11 c) \), and must prove \( 31 \mid(21 a+17 b+9 c) \) and \( 31 \mid(6 a+27 b+7 c) \), we frame these challenges as logical sequences. By strategically rearranging and relating terms in the given expression, we illustrate that if one condition holds, others do as well. Attention to these proofs doesn't just certify correctness but deepens understanding of the interconnectedness of mathematical relationships.
  • Mathematical proofs validate statements using logical steps from given premises to conclusions.
  • In proofs involving divisibility, rearranging terms is a common technique to demonstrate equivalent outcomes.

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Most popular questions from this chapter

Given \(r \in \mathbf{Z}^{+}\), write \(r=r_{0}+r_{1} \cdot 10+r_{2} \cdot 10^{2}+\cdots+\) \(r_{n} \cdot 10^{n}\), where \(0 \leq r_{i} \leq 9\) for \(1 \leq i \leq n-1\), and \(0<\) \(r_{n} \leq 9 .\) a) Prove that \(9 \mid r\) if and only if \(9 \mid\left(r_{n}+r_{n-1}+\cdots+\right.\) \(\left.r_{2}+r_{1}+r_{0}\right)\). b) Prove that \(3 \mid r\) if and only if \(3 \mid\left(r_{n}+r_{n-1}+\cdots+\right.\) \(\left.r_{2}+r_{1}+r_{0}\right)\). c) If \(t=137486 x 225\), where \(x\) is a single digit, determine the value(s) of \(x\) such that \(3 \mid t\). Which values of \(x\) make \(t\) divisible by 9 ?

For \(a, b \in \mathbf{Z}^{+}\)and \(s, t \in \mathbf{Z}\), what can we say about \(\operatorname{gcd}(a, b)\) if a) \(a s+b t=2\) ? b) \(a s+b t=3\) ? c) \(a s+b t=4\) ? d) \(a s+b t=6\) ?

As in Example \(4.18\) let \(L_{0}, L_{1}, L_{2}, \ldots\) denote the Lucas numbers, where (1) \(L_{0}=2\), \(L_{1}=1\); and, (2) \(L_{n+2}=L_{n+1}+L_{n}\), for \(n \geq 0\). When \(n \geq 1\), prove that $$ L_{1}^{2}+L_{2}^{2}+L_{3}^{2}+\cdots+L_{n}^{2}=L_{n} L_{n+1}-2 $$

Two hundred coins numbered 1 to 200 are put in a row across the top of a cafeteria table. Two hundred students are assigned numbers (from 1 to 200 ) and are asked to turn over certain coins. The student assigned number 1 is supposed to turn over all the coins. The student assigned number 2 is supposed to turn over every other coin, starting with the second coin. In general, the student assigned the number \(n\), for each \(1 \leq n \leq 200\), is supposed to turn over every \(n\)th coin, starting with the \(n\)th coin. a) How many times will the 200 th coin be turned over? b) Will any other coin(s) be turned over as many times as the 200 th coin? c) Will any coin be turned over more times than the 200 th coin?

Let \(a, d\) be fixed integers. Determine a summation formula for \(a+(a+d)+(a+2 d)+\cdots+(a+\) \((n-1) d)\), for \(n \in \mathbf{Z}^{*}\). Verify your result by mathematical induction.
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