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Chapter 7: Problem 20

If repeated divisions by 20,483 are performed, how many distinct remainders can be obtained?

Short Answer

Expert verified
Since the divisor 20,483 is a prime number and there are no factors between 1 and 20,482 that evenly divide it, all the remainders will be distinct. Therefore, there are \(20,483\) distinct remainders possible when dividing by 20,483.

Step by step solution

01

Identify possible remainders

When dividing by a number, the possible remainders range from 0 to one less than the divisor. Therefore, there can be remainders from 0 to 20,482 when dividing by 20,483.
02

Determine distinct remainders

Since each division results in a remainder from 0 to 20,482, there are a total of 20,483 possible remainders (including 0). However, to determine how many of these remainders are distinct, we need to make sure that none of them are the same.
03

Analyze the divisor

Observe the divisor, 20,483. As it is a prime number, there will be no factor between 1 and 20,482 that evenly divides it. Thus, when dividing any of these numbers by 20,483, it will yield a distinct remainder.
04

Conclusion

Since the divisor is a prime number and there are no factors that evenly divide it, all the remainders will be distinct. Therefore, there are 20,483 distinct remainders possible when dividing by 20,483.

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Key Concepts

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

Prime Numbers
Prime numbers are unique and fascinating elements in mathematics. They are defined as numbers greater than 1 that have no divisors other than 1 and themselves. This means a prime number cannot be divided evenly by any other number, which makes divisors for such numbers rare.
  • Examples include 2, 3, 5, 7, 11, and many others.
  • Prime numbers play critical roles in various fields, and one famous use is in encryption algorithms because of their divisibility properties.
For any given prime number like 20,483, a key understanding is that it's only divisible by 1 and itself. This implies that in operations like division, the characteristics of having no factor result in particular behavior related to remainders. Understanding this can help determine the outcomes of division problems and identify possible remainders.
Division Algorithm
The division algorithm is a fundamental tool in arithmetic and algebra, describing the outcome of division in terms of a quotient and a remainder. It states that for any two integers, a dividend \(a\) and a non-zero divisor \(b\), there exist unique integers \(q\) (quotient) and \(r\) (remainder) such that:
  • \(a = bq + r\)
  • where \(0 \leq r < b\).
This means when you divide a number by another, you are essentially finding how many times the divisor can "fit" into the dividend and what's left over. This leftover part is the remainder. The division algorithm perfectly explains why in any division by 20,483, the remainders could range from 0 to 20,482. Understanding this concept helps one grasp why there are 20,483 distinct remainders possible when dividing by a given prime number.
Remainders
Remainders are a crucial part of division, indicating what is left after the division of one number by another. When you divide a number (dividend) by another number (divisor), the remainder is what's left that cannot be evenly divided by the divisor.
  • For instance, dividing 10 by 3 gives a remainder of 1, since 3 fits into 10 three times (9), with 1 left over.
  • In division - 10 divided by 3 - you get a quotient of 3 and a remainder of 1.
In terms of mathematical properties, knowledge of prime numbers impacts how remainders behave when dividing. For the divisor, 20,483, a prime, every integer from 1 to 20,482 ensures a different remainder due to the absence of divisors that result in zeroing out the division, reflecting the prime's unique qualities. Thus, every operation with this divisor yields a distinct remainder.

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Most popular questions from this chapter

Student \(\mathrm{C}\) tries to define a function \(h: \mathbf{Q} \rightarrow \mathbf{Q}\) by the rule \(h\left(\frac{m}{n}\right)=\frac{m^{2}}{n}\), for all integers \(m\) and \(n\) with \(n \neq 0\). Student D claims that \(h\) is not well defined. Justify student D's claim.

Let \(S\) be the set of all strings of 0 's and 1 's, and define \(l: S \rightarrow \mathbf{Z}^{\text {nonneg }}\) by \(l(s)=\) the length of \(s\), for all strings \(s\) in \(S .\) a. Is \(l\) one-to-one? Prove or give a counterexample. b. Is \(l\) onto? Prove or give a counterexample.

Define \(F: \mathbf{Z}^{+} \times \mathbf{Z}^{+} \rightarrow \mathbf{Z}^{+}\)and \(G: \mathbf{Z}^{+} \times \mathbf{Z}^{+} \rightarrow \mathbf{Z}^{+}\)as follows: for all \((n, m) \in \mathbf{Z}^{+} \times \mathbf{Z}^{+}\), $$ F(n, m)=3^{n} 5^{m} \quad \text { and } \quad G(n, m)=3^{n} 6^{m} . $$ a. Is \(F\) one-to-one? Prove or give a counterexample. b. Is \(G\) one-to-one? Prove or give a counterexample.

Define \(F: \mathbf{R} \rightarrow \mathbf{R}\) and \(G: \mathbf{R} \rightarrow \mathbf{Z}\) by the following formulas: \(F(x)=x^{2} / 3\) and \(G(x)=\lfloor x\rfloor\) for all \(x \in \mathbf{R}\). What is \((G \circ F)(2) ?(G \circ F)(-3) ?(G \circ F)(5) ?\)

a. Suppose \(a_{1}, a_{2} \ldots \ldots, a_{n}\) is a sequence of \(n\) integers none of which is divisible by \(n\). Show that at least one of the differences \(a_{i}-a_{j}\) (for \(i \neq j\) ) must be divisible by \(n\). \(\boldsymbol{H}\) b. Show that every finite sequence \(x_{1}, x_{2}, \ldots, x_{n}\) of \(n\) integers has a consecutive subsequence \(x_{l+1}, x_{i+2}, \ldots, x_{j}\) whose sum is divisible by \(n\). (For instance, the sequence \(3,4,17,7,16\) has the consecutive subsequence \(17,7,16\) whose sum is divisible by \(5 .)^{*}\)
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