If a is an integer such that: 2 ła, 5ła Show that (a)⁴ⁿ = 1(mod 40) for any positive integer n

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: a) Prove that if p is prime and p  1≡ (mod 4), then ((p-1)/2)!)^2 ≡ -1 (mod p) b) Use part a) to…

A: According to the given information in part (a) it is required to prove that:

Q: Let m be fixed positive integer. Then an integers a is congruent to an integer b modulo m, if: ab|m…

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Q: Prove that 01 + 02 is even.

A: Any odd integer, p can be represented as p=2k+1 . Any even integer, q can be represented as q=2k .…

Q: Let a, b be integers. What does it mean for a = b (mod 17)? Describe the elements of Z17. Compute…

A: a ≡ b (mod 17) means that a − b = 17k for some k ∈ ℤ. ∴  b − a = −17k = 17(−k); henceb ≡ a (mod…

Q: Let a, b, n be positive integers. Prove that if a = b (mod n), then GCD(a, n) = GCD(b,n).

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Q: If n2 4 is an integer,prove by induction that n<nl.

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Q: Fermat's "Little" Theorem states that whenver n is prime and a is an integer, a"-l = 1 = 1 mod n a)…

A: Given a =463 and  n=953 We need to find x such that                   463952 ≡ x (mod 953).

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: For p an odd prime in Z , Prove: If p=(a)^2 + (b)^2  is possible in Z then   p= 1 (mod 4)

A: Compute general prime that satisfies given condition as follows.

Q: |Let m be a positive integer. Let a and b be integers such that a = b (modm). Use mathematical…

A: Solution

Q: (D) Let s be an integer such that 1 ss < 104651 and 24578° mod 104651 = 3190 Find s and explain how…

A: Let s be the integer such that and Purpose: To find the value of s

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: For any integer a, prove that a = 0 or 1 (mod 3).

A: Every number is either of the forms 3k, 3k−1, 3k+1 where k is an integer

Q: a) Find integers a and b such that a + b =a-b (mod 5).

A:

Q: a) Prove that 10" =+1 (mod 11) for any n EN. b) Suppose the integer r has digits rI-1I, I0. Prove…

A: Want to prove:(a) 10n≡±1 (mod 11) for any n∈N(b) If the integer x has digits xnxn-1.....x1x0 thenx≡0…

Q: Prove that for a positive integer n we have that 33n+3 – 26n – 27 is divisible by 169.

A: Topic - principle of mathematical induction

Q: Let n be a positive integer, and a an integer Prove that if the order of a modulon is n-1 then n is…

A: The given problem is to prove that n is prime from the given problem given a modulo n is set of…

Q: Find an integer a such that a 400 = 1(mod 3911).

A:

Q: 13. (a) Prove that for each integer a, if a 0 (mod 7), then a #0 (mod 7).

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Q: 5. Let a, b e Z and n E Z*. Prove that if a = b(mod n), then gcd(a, n) = gcd(b, n). (Hint: use the…

A: The greatest common divisor (gcd) of two or more numbers is the greatest common factor number that…

Q: Use the QRT to explain what ' 35 mod 4' means. There exists a unique integer r such that 3514 There…

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Q: Establish that if a is an odd integer, then for any n ≥ 1 a 2 n ≡ 1 (mod 2n+2). [Hint: Proceed…

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Q: Show that for any positive integer x, at least one of x, x+ 2, or x+ 4 is divisible by 3

A: The objective is to show that for any positive integer x at one of x, x+ 2, or x+ 4 is divisible by…

Q: Establish that if a is an odd integer, then 2" а = 1 (mod 2" +2) for any n 2 1.

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Q: Let m be an integer, if m is an even integer then prove that m + 7 is odd

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Q: Let n = 2p for some prime number p &gt; 2. Show that if a is a positive integer such that gcd(n , a)…

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Q: If m and n are relatively prime integers, then: a) mº(n) + nº(m) = 1 mod(mn) b) т+nmod (mn) c) -1…

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Q: Let m1, m2,…, mn be pairwise relatively prime integers greater than or equal to 2. Show that if a ≡…

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Q: 16. Prove that an integer (a,-1an-2 · . . ao)10 is divisible by 11 if and only if ao + a2 + a4 + . .…

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Q: Let p > 3 be a prime. Show that if p2+ p + 1 is also prime, then p =5 mod 6.

A: We prove the result by the method of contradiction.

Q: Prove that if a1, a2, . . a, are n > 2 integers such that a; = 1 (mod 3) for every integer i (1 < i<…

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Q: 15. Prove that for each natural number n, s+ (4n-3)(4n+1) 4n+1 ...

A: Given The given expression is 11·5+15·9+·  ·  ·+14n-34n+1=n4n+1

Q: Let a be an integer and let n e N. (a) Prove that if a = 0 (mod n), then n | a.

A: By the definition if congruence let a, b and n are integer, with n &gt;0, then a is congruent to b…

Q: Suppose n is an integer. Prove that if n is even, then n²+3n+5 is odd.

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Q: Let m be fixed positive integer. Then an integers ais congruent to an integer b modulom, if: a-b|m…

A: Consider the given information:

Q: Consider numbers of the form 9" mod 13, where n E Z+. We claim that all of these numbers will only…

A: First principle of mathematical induction : Suppose Pn is a statement where n∈ℤ+ . The statement is…

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

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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: (b) Let x be an integer such that 0 < x < 8 and (10500 + 105000 ) = x (mod 9). Justifying your…

A: We have to find the required value of x .

Q: Find the smallest positive integer k satisfying the stated condition or explain no such k exists:…

A: To find the smallest positive integer k satisfying the given condition of congruence.

Q: show that If n is a positive integer and a is any integer prime to n, then a ln) = 1(mod n), where…

A: It's called Euler's generalization to Fermats theorem. I have used integer m instead of n.

Q: Prove that if n is an odd positive integer, then n² = 1 (mod 8).

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Q: Let n E Z. Prove that if 5n2 +3 =0 (mod 8), then 4 n.

A: In this question  we will be using contradiction concept so assume it is true statement then we…

Q: Let a E Z. Prove that if a = 2 (mod 6), then a? = 4 (mod 12).

A: We will prove this by basic result of congruence

Q: Q2/let n be a fixed positive integer and a,b be arbitrary integers. Then a = b (modn) iffa and b…

A: We have to prove that Given statement

Q: 2.1.3. If a, b are integers such that a = b (mod p) for every positive prime p, prove that a = b.

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

A: Given:                 n=195 To find:                an-2=? by using Fermat's theorem Fermat's…

Q: Show that if a is an even integer, then a =0 (mod 4), and if a is an odd integer, then a =1 (mod 4).

A: Suppose a is an even integer Then a=2m for some integer m a2=(2m)2 a2=4m2Clearly 4|4so 4|4m2so…

Q: Prove that for any integer a, at = 0 (mod 5) or aª = 1 (mod 5)

A: To prove for any integer a,  a4≡0mod5 or a4≡1mod5.

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.