If n| m and a = b(mod m), then a = b(mod n).

Q: 6. Let m, n EN with m, n > 2, and let a, b E Z. Prove that a = b (mod mn) implies a = b (mod m) and…

A: (a) a = b (mod mn)       => there exists some integer k such that                    a = b +…

Q: Let n E Z+, n > 2. Show that n is prime if and only if (n – 2)! = 1 (mod n).

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Q: Prove that if a = b (modn) and m/n, then a = b (modm).

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Q: Prove that : If p is a prime number and p does not divide a then a- 1(modp).

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Q: if a equiv b (mod n) , then a⁴ equiv b⁴ (mod n) for any positive integer 4.

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Q: If a = b (mod m) and a = b (mod n) then a = b (mod mn)

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Q: Let a, be Z and n E N. For each statement decide whether it is necessarily true, or whether it can…

A:

Q: Prove for any integers a and b, if (a3+b3) mod 3 is not 0, then (a+b) mod 3 is not 0. (Hint: recall…

A: Congruence Modulo Relation on Integers: Let a, b∈ℤ and let m be a fixed positive integer.  Then a…

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: If a≡b (mod m) and c ≡d (mod m),show that (b) ac≡bd (mod m).

A: Given : a≡b (mod m) c ≡d (mod m), We have to show  ac≡bd (mod m)

Q: If a = b (mod n) and a = c (mod n) then which of the following is true. (a) b = c (mod n) (b) b2 =…

A: We use the properties of congruences to answer the given question.

Q: Suppose a and b are integers, a = 15 (mod 23), andb=7 (mod 23). Then integer c with Osc< 22 such…

A:

Q: Use the table in VII to give a formal proof of the following. For all integers a and b, if a = 1…

A:

Q: (c) For each integer a, if a 4 (mod 8), then a 2 (mod 8).

A:

Q: 1. Let a, b e Z. Prove that if a = 0 (mod 5) and b = 2 (mod 5), then a² + b² = 4 (mod 5).

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Q: If a =b (mod m) and a = b (mod n) then a = b (mod m+n) Select one: O True O False

A:

Q: Let a, b E Z and n E N. For each statement decide whether it is necessarily true, or whether it can…

A: The given problem is to prove the following identities, we have to use basic identities of modular…

Q: Prove or disprove If ac = bc(mod m), then a = b(mod m). If n| m and a = b(mod m), then a = b(mod n).…

A: Hello. Since you have posted multiple questions and not specified which question needs to be solved,…

Q: 5- If a = b mod (n), then ac = bc mod (n). 6- If a = b mod(n), then ak = bkmod(n) for every positive…

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Q: Let a, b e Z and n E Z*. Prove that if a = b(mod n), then gcd(a, n) = gcd(b, n).

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Q: For all integers a and b, if a = 7 ( mod 8)and b = 3 ( mod 8), then (a + b) = 2( mod 8).

A:

Q: If a = 1 (mod n) and b = 1 (mod n) then a - b = 0 (mod n) اختر واحدا: 2 3

A: True.

Q: If n is an integer, then (2n² + 3n)# 3(mod5).

A:

Q: 20. Are the following statements true or false? Either prove the statement is true or provide a…

A: PART A For the integers a and b, a.b=0(mod 6) This implies that either a=0(mod6) or b=0(mod6). So,…

Q: Let p be prime. (a) If a 1 (mod p), show that 1+ a + ...+ aP-1 = 1 (mod p). (b) If a -1 (mod p),…

A:

Q: (c) For all integers a and b, if a · b = 1 (mod 6), then a =1 (mod 6) or b = 1 (mod 6). (d) For all…

A: (c) The statement is false. For example take a=5, b=5 ab=2525≡1 mod(6) Hence ab≡1 mod(6) But a≡5…

Q: Ha = b (mod n) and gcd (a,n)= d, then gcd (b,n) %3D

A: Given a≡b(mod n). This can be written as a-b≡0(mod n). Implies a-b=kn for some natural number k.…

Q: Prove that min (a) If a = b (mod n) and mln ,then a = b (mod m) (b) If a = b (mod n) and c =d (mod…

A:

Q: Suppose that a and b are integers, a = 4(mod 13), and b = 9(mod 13). Find the integer c with 0<c< 12…

A: We will use the basic properties of congruence modulo n relation to answer this question.

Q: Suppose that each of a, b, and c is an integer and m is a positive integer. Show that if a = b(mod…

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Q: If p and q are distinct primes and a is an integer such that aº = a(modq) and aª = a(modp), then

A: This is a problem of theory of congruence, number theory.

Q: 23. Let a,be Z and n eN. If a =b (mod n), then a = ab (mod n). (24, If a b (mod n) and c=d (mod n),…

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Q: Prove directly from the definition of congruence modulo n that if a, c, and n are integers, n > 1,…

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Q: Let a,b,and c be natural numbers and n be a natural number > 0. Prove that if a = b mod n, then a^c…

A: Use mathematical induction to prove the given statement.

Q: Let n be an integer. Prove that, if n mod6 = 2, then n2mod12=4

A: We apply concept of modulus here to solve this.

Q: Give an example to show that a ^2 ≡ b^2 (mod m) need not imply that a≡ b (mod m)

A: Here I listed given example of above statement.

Q: Write a formal proof for the following. For all integers a and b, if a ≡ 1 (mod 4) and b ≡ 3 (mod…

A:

Q: If a = 3 (mod12) then a does not equal 1 (mod6)

A: a=b mod n if and only if n divides a-b Now, we have that a=3mod 12 We need to show that a≠1mod 6 Let…

Q: Prove the following:   a. If n is an integer and n + 3 is even, then n is odd by using a proof by…

A: Remark: Proof by contrapositive is based on the fact that an implication is equivalent to its…

Q: For any integer n, either n2 mod 4 = 0, or n² mod 4 = 1.

A: For any integer n, we prove that either n2 mod4=0,  or n2 mod4=1

Q: 20. If a e Z and a = 1 (mod 5), then a? = 1 (mod 5).

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Q: Let a, b, and c be integers and n a natural number. If a ≡ b mod n then a+c ≡ b+c mod n. How do I…

A:

Q: If ab = 0 (mod m), then either a = 0 (mod m) or D=0 (mod m). Select one: O True False

A:

Q: If a and b are integers such that a? = b² (mod p), where p is prime integer, then a) a = b (mod p)…

A:

Q: Prove or disprove: for any integers a, b, c, d, if a= c (mod n) and b= d (mod n), then ab = cd (mod…

A: Given: For any integers a, b, c, d if a≡cmod n and b≡dmod n To show: ab≡cd (mod n) To say that a is…

Q: For n=195, by Fermat's theorem a-2 a) a' (modn) b) 1(modn) c) 0(modn) d) a(mod n)

A: Fermat's theorem

Q: (a) Consider the following statement: For all n € N, x(x − 1) = 0 mod n Which of "if", "only if",…

A:

Q: 20. If a e Z and a = 1 (mod 5), then a² = 1 (mod 5). If a - h (mod n) then a3 = h3 (mod n)

A: Notation: a|b  means a divides b or                   a|b means b is divisible by a.

Q: 1 Prove each of the following assertions: (a If a = b (mod n) and m |n, then a = b (mod m). (6) If a…

A: We know that  a ≡ b (mod n) is equivalent to n | (b-a)

Q: Let n be a positive integer, and assume that a and b are integers such that a ≡ b (mod n). Prove…

A: Looking at the definition of congurence we can easily do these two problems and we need some algebra…

Coordinate Geometry

Coordinate geometry, also known as analytic geometry or Cartesian geometry in classical mathematics, is a type of geometry that is studied using a coordinate system. Coordinate geometry is a branch of mathematics that describes the position of points on a plane using an ordered pair of numbers called coordinates.