Prove that there is no integer n such that n ≡ 4 (mod 10) and n ≡ 3 (mod 15)

Q: 8. Let m 2q + 1 be an odd positive integer. Prove that every integer = is congruent modulo m to a…

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Q: Find the smallest positive integer k such that 3k = 1 (mod 19).

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Q: Show that if p is an odd prime, then 2(p - 3)!= -1 (mod p) Prove that if n has an odd number of…

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Q: Find an integer n such that n = 1 mod 7, n = 3 mod 8, and n = 2 mod 9.

A: We can solve this system of congruences using the Chinese Remainder Theorem.

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

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Q: Suppose that 2n−1 ≡ 1 (mod n). Is n necessarily prime?

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Q: that there does not exist an integer n such that n= 21),

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Q: Show that if p is an odd prime and a and b are integers with a ≡ b (mod p), then

A: According to the given information, let p be an odd prime and a and b are integers with a = b…

Q: Prove that 2 is not a primitive root modulo 31.

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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: 3. Prove: for every integer n and for every odd integer a² = 1 (mod 2n+2)

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Q: Prove or disprove by using "congruence modulo n" and "divides." (a) For all positive integers a, x…

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Q: Let p be a prime with p ≡ 3 (mod 4), and suppose that q = 2q + 1 is also prime. Determine if 2 is a…

A: let p be a prime with p≡3mod 4 and suppose that q=2p+1 is also prime determine if 2 is a square mod…

Q: Solve the congruence 11x = 45 (mod 97) where x is an integer in the interval [0,97).

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Q: Prove that if x² = y² (mod n) and x #±y (mod n) then gcd(x + y, n) is a nontrivial factor of n.

A: We have,x2≡y2mod nAnd,x≢±ymod n. We have to show that gcdx+y, n is a nontrivial factor of n.

Q: Prove that an odd integer n is a square modulo every power of 2 if and only if n ≡8 1.

A: Given n is an odd integer. We have to prove that an odd integer n is a square modulo every power of…

Q: 11. Prove that a¹ = 1 (mod 5) for every integer a that is not divisible by 5.

A: We have to prove  for every integer  that is not divisible by

Q: 2

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Q: Characterize the primes p mod 20 such ial

A: We have to characterize p mod 20 such that -5p=1.

Q: DuppOse that a anа т аге intege Such that a, аre al diпerent IMod- ulo m, with am-1 = 1 mod m. Show…

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Q: Find all integers between 0 and 20 that are a perfect squares of mod 21

A: 4, 9, 16 only three numbers between 0 and 20 whose perfect squares are integers under mod 21 1) √4…

Q: Prove for each integer n, that n2 is congruent either to 0 or to 1 modulo 4.

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Q: Prove that for any integer x, if x mod 3 does not divide 0, then (x2 −3x + 1) mod 3 = 2.

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Q: 2) Find the smallest positive integer z, such that z = 13 mod 4 z = 20 mod 9 z = 11 mod 25 .

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Q: Let 1 ≤k≤p-1 where p is a prime number. Then prove that (₁7¹) = (-1) (modp) k

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Q: 4) Determine if the following statement is true and either prove it or provide a counter example:…

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Q: Show that the Bernoulli number B₁ satisfies B₁ = (-1)" ( 2 + 2d²+1) Σ (mod 1) for any positive…

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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: For every integer n &gt; 1, prove that (n - 1)!mod n = n - 1 if and only if n is prime.

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Q: (b) Give an example of a composite integer n such that the congruence x =1 (mod n) pas more than two…

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Q: a) Let a be an integer with a > 0. The integer a is even if and only if a = 0 mod 2. Note to self:…

A: (a) Let a be an integer with a &gt; 0.       We need to prove that a is even if and only if a ≡ 0…

Q: Let n1, n2, n3 be three numbers that are pairwise relatively prime so that (n1, n2) = (n2, n3) =…

A: Given that n1,n2 and n3 are three numbers that are pairwise relatively prime. We have to prove…

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: 1. Let a and b be natural numbers, where a, b > 2, such that gcd(a, b) = 1 and ab = (x + y)" for…

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Q: 25 Prove that mod 7 5 for all Gindegars and bin b mod 7-6

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Q: Find a two digit integer c such that 101c = 1 mod 107. This congruence has a unique solution mod 101…

A: We need to find a digit integer c such that 101c ≡ 1 mod 107. As per the given question, this…

Q: Find all integers X satisfying X ≡ 1 mod 6, X ≡ 3 mod 8, X ≡ 7 mod 9

A: First divide 1 i.e. the dividend by 6 i.e. the divisor 1/6 = 0.16667 Then multiple the whole part of…

Q: 4. Let p> 2 be prime. Show that there exists an integer n such that n² + 1 is a multiple of p if and…

A: We use the fact that if gcd(a,p)=1 then a^{p-1}= 1 (mod p)