Division and Congruence The primary difference between modular and normal arithmetic is, per-haps unsurprisingly, with regard to division.Theorem 3.9. If ka = kb (mod kn) then a = b (mod n).The modulus is divided by k as well as the terms, so the meaning of = changes. In Exercise 3.1.12you will prove this theorem, and observe that, in general, we do not expect a = b (mod n). 3.1.12 Suppose that 7x = 28 (mod 42). By Theorem 3.9, it follows that x = 4 (mod 6).(a) Check this explicitly using Theorem 3.6.(b) If 7x = 28 (mod 42), is it possible that x = 4 (mod 42)?(c) Is it always the case that 7x = 28 (mod 42) → x = 4 (mod 42)? Why/why not?(d) Prove Theorem 3.9.

Answered: Image /qna-images/answer/7bcb3bd8-65fa-4725-84f7-0924f3f6ad78.jpg

3.1.12 Suppose that 7x = 28 (mod 42). By Theorem 3.9, it follows that x = 4 (mod 6). (a) Check this explicitly using Theorem 3.6. (b) If 7x = 28 (mod 42), is it possible that x = 4 (mod 42)? (c) Is it always the case that 7x = 28 (mod 42) → x = 4 (mod 42)? Why/why not? (d) Prove Theorem 3.9.

3.1.12 Suppose that 7x = 28 (mod 42). By Theorem 3.9, it follows that x = 4 (mod 6). (a) Check this explicitly using Theorem 3.6. (b) If 7x = 28 (mod 42), is it possible that x = 4 (mod 42)? (c) Is it always the case that 7x = 28 (mod 42) → x = 4 (mod 42)? Why/why not? (d) Prove Theorem 3.9.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: k) Prove that x = 5108 = 179 (mod 433) is a solution to the equation x² + 1 = 0 (mod 433), and use…

A: 52 ≡25(mod 433)54 ≡625≡192(mod 433)55 ≡960≡94(mod 433)56 ≡470≡37(mod 433)512 ≡1369≡70(mod 433)524…

Q: 2-) Find all in tegers x such thet 5x = 12 (mod 7) 10x = 4 (mod 1) 7x E 16 (mod 10d ond 1occ < x <…

A: Given: 5x≡12(mod 7) 10x≡4(mod11) 7x≡16(mod 10) and 1000≤x≤3000

Q: 9. Let a and b be integers. Prove that if a =7 (mod 8) and b = 3 (mod 8), then: (a) a + b = 2 (mod…

A: Your questions solution below in images

Q: Compute 9-¹1 (mod 19) using Fermat's Little Theorem. Show your work.

Q: 2. What is the least residue of 610 C 512 10.4512 (mad 1112

A: Euler's theorem is used

Q: If x = y (mod n) and u = v (mod n) then which of the following is false. (a) x+ u = y +v (mod n) (b)…

A: Given that x≡ymod n and u≡v mod n. By definition, x≡ymod n implies x-y is divisible by n. Then let,…

Q: For each statement, indicate whether it is true or false. Then prove or disprove it. (a) For all SC…

A: (a) FALSE. Consider S ⊆ℤ, defined by S = { 2+8k | k ∈ℤ } U { 2+12r | r∈ℤ }. Then the given…

Q: The remainder when a is divided by 7 is 2 and the remainder when b is divided by 7 is 5. (So a mod 7…

A: Given that remainder when "a" is divided by 7 is 2

Q: Show that (p-1)(p2) (pr) (-1)'r!(mod p), for r= 1, 2, . . . ,p - 1. .

Q: Suppose that a and b are integers such that a = 45 (mod 71) and b = 53 (mod 71). Find an integer c…

Q: If al = b (mod p), then prove that a = b (mod p).

A: (ii) given that ap≡bp (mod p) we have to prove…

Q: 5. Find an integer x between 0 and 776 such that x = 13 (mod 21) x = 5 (mod 37)

A: Set N = 21 × 37 = 777. Then , m1 = N/21 = 777/21 = 37 and m2 = N/37 = 777/37 = 21 Now we need to…

Q: 1. How many solutions( mod 419) does x = 186 mod 419 have? (Show your work!) 2. How many solutions (…

A: Given: x2≡186 mod 101 To find: Number of solutions that exists. Method Used: For any, xk≡b mod p,…

Q: If x = y (mod n) and u = v (mod n) then which of the following is true (a) x + u = y + v (mod n) (b)…

A: Given that x≡y (mod n) and u≡v (mod n)

Q: Suppose that a and b are integers. a ≡ 7 (mod 13) and b ≡ 9 (mod 13). Find the integer c such that 0…

A: The given problem is to find the modular values of c for the given conditions with given integers a…

Q: 5. Let f(x) = 2x² + x + 4. It is known that f(1) = 0(mod 7). (i). Find a solution to f(x) = 0(mod…

Q: If a = b (mod n) and b =c (mod n) then which of the following is false. a = c (mod n) a = c3(mod n)…

Q: 2. Prove that if ar b (mod n), then () x = () (mod ()) where d = gcd(a, n). Apply this result to 5x…

A: We first find the gcd of the given numbers and then use that to solve the given congruence. The…

Q: 4. Using your work from the earlier parts or independently find: (a) mod 8 (b) + mod 23 17 (c) mod 7

A: Given

Q: (b) By applying part (a), solve the congruences 2x = 1 (mod 31), 6x = 5 (mod 11) 3x = 17 (mod 29).…

Q: If a≡3 (mod 11) and b≡9 (mod 11): (a) Compute a+b(mod 11) (b) Compute a^3(mod 11)

Q: Suppose x satisfies the three simultaneous congruences X = 4 (mod 5) x = 7 (mod 8) X 10 (mod 11)…

Q: *1. What is the least residue of 56 (mod 7) 58 (mod 7) 19458 (mod 7)?

A: Note that : Since you have posted multiple questions, we will provide the solution only to the first…

Q: If x00 = 1(mod 109) then possible value of x is a) 108 (b) 218 (c) 109 (d) None of these

Q: 13. In Z26, it is true that 3-9 = 1, 5-21 = 1, 7-15 = 1, 11-19 = 1, and 17-23 = 1 mod 26. Use this…

A: Modulo is defined as the remainder getting when one number is divided by another number. Modulo is…

Q: Find the first few terms of the sequence of pseudorandom numbers generated using the linear…

Q: Which of the following are true modulo 12? (1) Does 5z = 6 mod 12 have a solution? How many? (2)…

Q: Evaluate the following: 495 (mod 11) 3(mod 8) 2345(mod 6) (i) (ii) (iii) 9

A: As per bartleby guidelines we only give first three questions answers. Please repost for other…

Q: Find the first few terms of the sequence of pseudorandom numbers generated using the linear…

Q: Compute 21200 mod 27.

A: We see, 21 ≡ (-6) mod 27 => 212 ≡ (-6)2 mod 27 ≡ 36 mod 27…

Q: 11.5. For each part, find an x that solves the given simultaneous congruences. (a) x = 3 (mod 7) and…

Q: Suppose that a = 3 mod 12 and b =7 mod 12 IfC = 3 a – 2 b mod 12 then C =

Q: If 19 =7 (mod 6) and 27 =3 (mod 6) then 19x 27 = 7+3 (mod 6) 19+27 = 7+3 (mod 6) 19+27 =7x 3 (mod 6)…

Q: 13. Suppose that z1 V3 +i and z2 = 2/2+ 2v2i. 22 (a) Write z1 and z2 in mod-arg form. (b) Hence…

Q: Let a € Z be odd. (a) Prove that a = 1 (mod 4) or a = 3 (mod 4). (b) Prove that a² = 1 (mod 4).

Q: 2. Find the largest negative x € Z that simultancously satisfies 4x = 9 (mod 5) 2x = 0 (mod 6) x =0…

A: Given : 4x = 9 (mod 5) 2x = 0 (mod 6) x = 0 (mod 4)

Q: If a = b (mod n) and b = c (mod n) then which of the following is false. a =c (mod n) a = c (mod n)…

Q: If a = b (mod n) and b = c (mod n) then which of the following is true. (a) a = c (mod n) (b) a? =…

Q: Show that (-3) = 1 if q = ±1 mod 12 -1 if q = ±5 mod 12

Q: Need help with True or False for the following... (I will put what I believe As (T) or (F)) 1. For…

A: Let's evaluate the provided statements and their negations as well as the converses and…

Q: duce the expression for u to 2(5 ,Y = (1+5P) +4 (mod 2p - 1) or simply, to 2u, = 1+5P-1(mod 2p - 1).…

Q: If n 2 1, then the Fermat number Fn 22" +1 is prime if and only if %3D 3(F.-1)/2 = -1 mod (Fn).

Q: Work the modular arithmetic problem. (12 9)(mod 3). (12 9)(mod 3) =(mod 3)

A: The given modular arithmetic problem is - (12·9)(mod 3) We have to solve the above problem.

Q: 2 (-3) - = (- (-1)² 8 = {) 1 -1 if p= ±1 (mod 8) if p= ±3 (mod 8).

Question

Division and Congruence The primary difference between modular and normal arithmetic is, per-
haps unsurprisingly, with regard to division.
Theorem 3.9. If ka = kb (mod kn) then a = b (mod n).
The modulus is divided by k as well as the terms, so the meaning of = changes. In Exercise 3.1.12
you will prove this theorem, and observe that, in general, we do not expect a = b (mod n).

3.1.12 Suppose that 7x = 28 (mod 42). By Theorem 3.9, it follows that x = 4 (mod 6).
(a) Check this explicitly using Theorem 3.6.
(b) If 7x = 28 (mod 42), is it possible that x = 4 (mod 42)?
(c) Is it always the case that 7x = 28 (mod 42) → x = 4 (mod 42)? Why/why not?
(d) Prove Theorem 3.9.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 2 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Ring$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

If x = y (mod n) and u = v (mod n) then which of the following is false. (a) x+ u= y + v (mod n) (b) xu= yv (mod n) (с) x+e=y+c (mod cn) (d) None of these. :Select one b C d a Clear my choice
Show that if p = 1 (mod 4), then x² = -1 (mod p) has a solution given by the least residue (mod p) of ((p-1)/2)!.
Let x, y ∈Z. Prove that if x≡1(mod 5) and y ≡2(mod5 ), then x^2-y^2≡0(mod 5). Give 2 examples to show that this scenario either does or does not work. Give the proof with an explanation to each step.
Refer to image
Prove that (50!)2 mod 101 = -1 mod 101.
11. Show that 2 (p - 3)! + 1 = 0 (mod p).
2. Solve the simultaneous congruences x = 4 (mod 5) and 2x = 3 (mod 7)
Please solve this question
1)For each of the following, find an integer value x which satisfies the equation, or explain why there is no solution to the equation. (a) 2x ≡ 3(mod 25) (b) 6x ≡ 4(mod 15)
4 ≡≡ (mod 8)
4) Find 25725 (mod 31)
How can I prove this?