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Chapter 4: Problem 36

Find each of these values. a) \((177 \bmod 31+270 \bmod 31) \bmod 31\) b) \((177 \bmod 31 \cdot 270 \bmod 31) \bmod 31\)

Short Answer

Expert verified
a) 13b) 19

Step by step solution

01

Calculate 177 mod 31

First, find the remainder when 177 is divided by 31. Calculate: 177 梅 31 = 5 remainder 22 So, 177 mod 31 = 22
02

Calculate 270 mod 31

Next, find the remainder when 270 is divided by 31. Calculate: 270 梅 31 鈮� 8 remainder 22 So, 270 mod 31 = 22
03

Add the results and take mod 31

Add the remainders calculated in Step 1 and Step 2 and then take modulo 31 of the result. 22 + 22 = 44 44 mod 31 = 13 So, (177 mod 31 + 270 mod 31) mod 31 = 13
04

Multiply the results and take mod 31

Multiply the remainders calculated in Step 1 and Step 2 and then take modulo 31 of the result. 22 * 22 = 484 484 mod 31 = 19 So, (177 mod 31 * 270 mod 31) mod 31 = 19

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Key Concepts

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

modulo operation
The modulo operation is a fundamental concept in modular arithmetic. It finds the remainder after division of one number by another. For example, in the expression \(177 \bmod 31\), 177 is divided by 31 to give a quotient and a remainder. The modulo operation returns this remainder. In this case, 177 divided by 31 equals 5 with a remainder of 22, so 177 mod 31 equals 22. This operation is useful in many areas of mathematics and computer science, such as cryptography and algorithm design.
remainder
The remainder is what is 'left over' after performing a division. It is a key part of the modulo operation. For example, calculating \(270 \bmod 31\) involves dividing 270 by 31. The quotient is the number of times 31 can fully fit into 270, which is 8 times, and the remainder is what is left, which is 22. Thus, 270 mod 31 equals 22. The remainder helps in understanding the distribution and grouping of numbers in modular systems.
modular addition
Modular addition is the process of adding two numbers within a modular system. First, calculate the remainders of each number mod a specific integer. Then, add these remainders together and apply the modulo operation again to the result. For example, to find \((177 \bmod 31 + 270 \bmod 31) \bmod 31\), first find 177 mod 31 (which is 22) and 270 mod 31 (also 22). Adding these gives 44. Then, 44 mod 31 equals 13. So, \((177 \bmod 31 + 270 \bmod 31) \bmod 31 = 13\). This operation helps keep the results within the bounds of the modulus.
modular multiplication
Modular multiplication involves multiplying numbers in a modular system. Calculate the remainders of each number mod a specific integer, multiply these remainders, and then apply the modulo operation to the product. For example, in solving \((177 \bmod 31 \times 270 \bmod 31) \bmod 31\), first find 177 mod 31 (which is 22) and 270 mod 31 (also 22). Multiplying these together gives 484. Then, 484 mod 31 equals 19. So, \((177 \bmod 31 \times 270 \bmod 31) \bmod 31 = 19\). This technique ensures the product remains within a manageable range.

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Most popular questions from this chapter

\(\begin{array}{ll}{\text { Find the Cantor expansions of }} \\ {\text { a) } 2 .} & {\text { b) } 7} \\ {\text { c) } 19 .} & {\text { d) } 87} \\ {\text { e) } 1000 .} & {\text { f) } 1,000,000}\end{array}\)

Determine whether the integers in each of these sets are pairwise relatively prime. $$\begin{array}{ll}{\text { a) } 21,34,55} & {\text { b) } 14,17,85} \\\ {\text { c) } 25,41,49,64} & {\text { d) } 17,18,19,23}\end{array}$$

a) Show that \(2^{340} \equiv 1(\bmod 11)\) by Fermat's little theorem and noting that \(2^{340}=\left(2^{10}\right)^{34} .\) b) Show that \(2^{340} \equiv 1(\bmod 31)\) using the fact that \(2^{340}=\) \(\left(2^{5}\right)^{68}=32^{68}\) . c) Conclude from parts (a) and (b) that \(2^{340} \equiv\) 1\((\bmod 341)\)

A parking lot has 31 visitor spaces, numbered from 0 to \(30 .\) Visitors are assigned parking spaces using the hashing function \(h(k)=k\) mod \(31,\) where \(k\) is the number formed from the first three digits on a visitor's license plate. a) Which spaces are assigned by the hashing function to cars that have these first three digits on their license plates: \(317,918,007,100,111,310 ?\) b) Describe a procedure visitors should follow to find a free parking space, when the space they are assigned is occupied. Another way to resolve collisions in hashing is to use double hashing. We use an initial hashing function \(h(k)=k \bmod p,\) where \(p\) is prime. We also use a second hashing function \(g(k)=(k+1) \bmod (p-2) .\) When a collision occurs, we use a probing sequence \(h(k, i)=(h(k)+i \cdot g(k)) \bmod p .\)

Show that the binary expansion of a positive integer can be obtained from its octal expansion by translating each octal digit into a block of three binary digits.
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