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Chapter 4: Problem 41

Show that if \(n | m,\) where \(n\) and \(m\) are integers greater than \(1,\) and if \(a \equiv b(\bmod m),\) where \(a\) and \(b\) are integers, then \(a \equiv b(\bmod n)\)

Short Answer

Expert verified
If \( n | m \) and \( a \equiv b (\bmod m) \), then \( a \equiv b (\bmod n) \) because \( a - b \) is divisible by both \( n \) and \( m \).

Step by step solution

01

Understand the Given Conditions

Given: 1) Integer division where \( n | m \), this means there exists an integer \( k \) such that \( m = nk \). 2) Congruence relation \( a \equiv b (\bmod m) \), which implies \( m | (a - b) \).
02

Express the Congruence Relation

From \( a \equiv b (\bmod m) \), it follows that there exists an integer \( q \) such that \( a - b = qm \).
03

Substitute \( m \) in Terms of \( n \)

Since \( m = nk \), substitute to get: \( a - b = q(nk) \). This simplifies to: \( a - b = (qk)n \).
04

Conclusion Using the Definition of Divisibility

Since \( a - b = (qk)n \), it implies \( n | (a - b) \), which means \( a \equiv b (\bmod n) \).

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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 key concept in number theory.
Here, we say that an integer \( n \) divides another integer \( m \) if there is an integer \( k \) such that \( m = nk \). This is written as \( n | m \).
For example, 3 divides 12 because 12 is equal to 3 times 4 (i.e., 12 = 3 * 4).
Understanding integer division is crucial because it helps us simplify problems and see relationships between numbers.
Congruence Relation
A congruence relation in number theory is a way of saying two numbers are equivalent when divided by a certain number.
We write that \( a \equiv b \ (\bmod \, m) \) if \( a - b \) is divisible by \( m \).
For example, if we have \( a = 14 \) and \( b = 8 \) under modulo \( m = 6 \), then \( a \equiv b \ (\bmod \, 6) \) because \( 14-8 = 6 \), which is divisible by 6.
In our exercise, we use the fact that if \( a \equiv b \ (\bmod \, m) \), then there is an integer \( q \) such that \( a - b = qm \). This relationship lets us transform equalities in modulo computation.
Divisibility
Divisibility is another fundamental concept in number theory.
It serves as a connection between different parts of math, especially in modular arithmetic.
If \( n \) divides \( m \) (written as \( n | m \)), it means \( m = nk \) for some integer \( k \). In the context of our exercise, this divisibility lets us simplify the modulus \( m \) into smaller parts.
For example, because \( m = nk \), and if \( m \) is a factor in a difference such as \( a - b \), then \( n \) must also be a factor.
Ultimately, understanding divisibility allows us to conclude that \( a \equiv b \ (\bmod \, n) \) from \( a \equiv b \ (\bmod \, m) \) if \( n | m \). This logical step concludes the proof.

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Most popular questions from this chapter

In Exercises \(31-32\) suppose that Alice and Bob have these public keys and corresponding private keys: \(\left(n_{\text { Alice }}, e_{\text { Alice }}\right)=\) \((2867,7)=(61 \cdot 47,7), \quad d_{\text { Alice }}=1183\) and \(\left(n_{\text { Bob }}, e_{\text { Bob }}\right)=\) \((3127,21)=(59 \cdot 53,21), d_{\text { Bob }}=1149 .\) First express your answers without carrying out the calculations. Then, using a computational aid, if available, perform the calculation to get the numerical answers. Alice wants to send to all her friends, including Bob, the message "SELL EVERYTHING" so that he knows that she sent it. What should she send to her friends, assuming she signs the message using the RSA cryptosystem.

Find \(\operatorname{gcd}(1000,625)\) and \(\operatorname{lcm}(1000,625)\) and verify that \(\operatorname{gcd}(1000,625) \cdot \operatorname{lcm}(1000,625)=1000 \cdot 625\)

In Exercises \(31-32\) suppose that Alice and Bob have these public keys and corresponding private keys: \(\left(n_{\text { Alice }}, e_{\text { Alice }}\right)=\) \((2867,7)=(61 \cdot 47,7), \quad d_{\text { Alice }}=1183\) and \(\left(n_{\text { Bob }}, e_{\text { Bob }}\right)=\) \((3127,21)=(59 \cdot 53,21), d_{\text { Bob }}=1149 .\) First express your answers without carrying out the calculations. Then, using a computational aid, if available, perform the calculation to get the numerical answers. Alice wants to send to Bob the message "BUY NOW" so that he knows that she sent it and so that only Bob can read it. What should she send to Bob, assuming she signs the message and then encrypts it using Bob's public key?

What is the value of \(\phi\left(p^{k}\right)\) when \(p\) is prime and \(k\) is a positive integer?

Suppose that the most common letter and the second most common letter in a long ciphertext produced by encrypting a plaintext using an affine cipher \(f(p)=(a p+\) \(b ) \bmod 26\) are \(Z\) and \(\mathrm{J}\) , respectively. What are the most likely values of \(a\) and \(b\) ?
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