Show that if p is a prime number then xy ≡ 0 mod p implies that x ≡ 0 mod p or y ≡ 0 mod p
Q: Suppose that p, q are distinct prime numbers, prove that p9-1 + gp-1 = 1 mod pq
A: We use Fermat's Little Theorem, to prove our claim.
Q: Prove that : If p is a prime number and p does not divide a then a- 1(modp).
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Q: Let p= 2k +1 be an odd prime number. Show that x* – 1 = (X – 1)(X – 2²) - · (X – k²) mod p.
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Q: Let p ≥ 2 be a prime. Suppose p does not divide a. Then ap−1 ≡ 1 (mod p).
A: The given problem is to show the given following modular congreunce that let p is a prime and p…
Q: 7. Find a whole number x in the interval [0, 30] that satisfies the following x = 1 mod 2 x = 2 mod…
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Q: consider the statement: for all n∈N , x∈Z , x^2 ≡ 1 mod n ( ) x ≡ 1 mod n or x≡ -1 mod n a)…
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Q: 24. Prove that for all integers m and n, if m mod 5 = 2 and n mod 5 = 1 then mn mod 5 = 2.
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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: Let n be a fixed positive integer greater than 1. If a mod n = a' andb mod n = b', prove that (a +…
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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: (i) Prove that if p is a prime, then (p - 1)! = -1 (mod p). (ii ) Show that 18! = -1 (mod 437).
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Q: Assuming Wilson's Theorem, prove that if p prime and (a, p) = 1, then aP-1 = 1 mod p.
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Q: Suppose r is a primitive root mod a prime p such that p = 3 mod 4. Prove ord,(-r) =
A: Given: p≡3mod 4 To find: ordpp-g=p-12
Q: Let P be a prime, if ac = bc (mod p) and p does not divide c, then a = b (mod p) Select one: O True…
A: Given gcd(c, p) = 1 and ac ≡ bc(mod p). (We may assume p > 1 so that c ≠ 0.) Then ac − bc =…
Q: Show that if a ≡ b (mod m) and c ≡ d (mod m), then a + c ≡ b + d (mod m).
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Q: Let a, b, c, m be integers with m > 2. Prove that if ac ≡ bc mod m and d = gcd(m, c) , then a ≡ b…
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Q: Suppose p is an odd prime number, prove that p+1 (р — 3)! %3D (mod p) -
A: To prove: To prove that p-3!≡-p+12(mod p) for an odd prime number p.
Q: 2. Prove that if a = b( mod r) and c = d( mod r) then a – c= b – d( mod r)
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Q: 1. Suppose m, nEz* are relatively prime. Prove that for all a, bEZ, a =b (mod mn) if a =b (mod m)…
A: Note that: a is congruent to b modulo m if and only if m divides (a-b).
Q: Show that if p and q are distinct primes, then pª-1 + q®=1 = 1 (mod pq).
A: Given problem is to show the following identity using congruence, we have to consider that…
Q: Prove that for any prime, p = 1 (mod 4) there are exactly (p-1)/4 incongruent values of a 0 (mod p)…
A: By theorem (a,p)=1 then xn≡a modp has g.c.d(n,p-1) solution or no solution according as…
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: Prove that for all integers x, if x is not divisible by 7, then x^3 ≡ 1 mod 7 or x^3 ≡−1 mod 7
A: We will prove the given statement.
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: 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…
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Q: mod n, then x ≡ 0 mod n or
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Q: Show that y = x (mod p) leads to -y = -x (mod p).
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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: Suppose that a1 ≡ b1 mod (m) and a2 ≡ b2 mod (m) -Show that a k 1 ≡ b k 1 mod (m) (hint: use…
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Q: (4) Let p be a prime number. Then r = 1 mod p has exactly two solutions modulo p.
A: 4) To prove, x2≡1mod p has exactly 2 roots such that p is a prime number.
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: Show that: if a = b(mod n) and C = d\modn), then a +c =b+d\mod n)!
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Q: Prove that (a mod 2)(b mod 2) = (ab) mod 2. ANY METHOD of proof.
A: Prove that a mod 2b mod 2=ab mod 2
Q: Prove that if a + x ≡ a +y (mod n), then x ≡y (mod n)
A: I have proved it by using defination of congruent.
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: Prove that if a = b mod n and c= d mod n, then ac bd mod n.
A: Given if a≡bmodn and c≡dmodn We have to show that ac≡bdmodn. Now, a≡bmodn…
Q: 2. Let p and q be distinct primes. Let a be an integer such that aº = a aª = a mod p. Prove that aPª…
A: ap = a (mod q) and aq = a (mod p)
Q: Prove that there is no integer n such that n ≡ 4 (mod 10) and n ≡ 3 (mod 15)
A: We have to prove that there is no integer n such that n ≡ 4 (mod 10) and n ≡ 3 (mod 15)
Q: Use a direct proof: p is a prime # and x ∈ Z. If x2≡x (mod p) then x ≡ 0 (mod p) or x ≡ 1 (mod p).…
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Q: Let p be prime. Suppose ab≡0 mod p and a ≢ 0 mod p. Apply Euclid’s Lemma to prove b ≡0 mod p
A: Given that, Let p be prime. Suppose, ab≡0 mod p and a≢0 mod p. To prove: b≡0 mod p…
Q: If ab = 0 (mod m), then either a = 0 (mod m) or D=0 (mod m). Select one: O True False
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Q: If a # 0 (mod m) and b # 0 (mod m), then ab# 0 (mod m). Select one: O True False
A: justification is given in step 2)
Q: Prove or disprove: If a = b(mod n²) then a = b(mod n)
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Q: Let P be a prime, if ac ≡ bc (mod p) and p does not divide c, then a ≡ b (mod p) Select one: True…
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Q: Prove the above property. That is, prove that if a mod n = k and b mod n = j, then (a + b) mod n =…
A: If a mod n =k and b mod n=j show that a+b mod n =j+k mod n. Definition: If x mod y=z, then y | x-z.…
Q: (2) If a = b modn and a = b mod m and ged(m, n): 1. then a =b mod mn.
A: The greatest common divisor of two or more integers, which are not all zero, is the largest positive…
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- 30. Prove that any positive integer is congruent to its units digit modulo .Prove that if a+xa+y(modn), then xy(modn).a. Prove that 10n(1)n(mod11) for every positive integer n. b. Prove that a positive integer z is divisible by 11 if and only if 11 divides a0-a1+a2-+(1)nan, when z is written in the form as described in the previous problem. a. Prove that 10n1(mod9) for every positive integer n. b. Prove that a positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9. (Hint: Any integer can be expressed in the form an10n+an110n1++a110+a0 where each ai is one of the digits 0,1,...,9.)
- In the congruences ax b (mod n) in Exercises 40-53, a and n may not be relatively prime. Use the results in Exercises 38 and 39 to determine whether these are solutions. If there are, find d incongruent solutions modulo n. 42x + 67 23 (mod 74)25. Complete the proof of Theorem : If and is any integer, then .In the congruences axb(modn) in Exercises 4053, a and n may not be relatively prime. Use the results in Exercises 38 and 39 to determine whether there are solutions. If there are, find d incongruent solutions modulo n. 8x66(mod78)