Show that: if a = b(mod n) and C = d\modn), then a +c =b+d\mod n)!

Q: Prove if a=b mod m then a^n=b^n mod m use Math induction

A: We have to prove that if a=b mod m then a^n=b^n mod m

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: Let a, m € Z+ and (a,m) = (a - 1, m) = 1. Then prove that 1+a+...+ a(m)-1 = 0(modm)

A: For given,       (a,m) = (a -1, m) = 1 We have to show,      1 + a + a 2 + ........ + a φ (m)-1 ≡ 0…

Q: Problem MCD = Greatest common divisor (GCD) t them.a, b, c and m €Zsuch that m 22. a-). Prove that a…

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Q: If cy (mod n) andc>0 then a) x= y (mod cn) b) x y (mod n) ) xy d) (mod") None of these

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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: If a cy (mod n) and c> 0 then a) x y (mod en) b) x y (mod n) ) x=y (mod) d) None of these :Select…

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Q: л There

A: We prove our result by using definition of mod.

Q: If a = b (mod n) and a = c (mod n) then which of the following is true. (a) b3 c (тod n) (b) b2 =…

A: Given below the detailed solution

Q: If a ≡ b (mod n1) and a ≡ b (mod n2), must it be true that a ≡ b (mod n1 + n2)? Either give a proof…

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Q: 1. Find all x, y, and z such that a) x = 3 (mod 4) x = 5 (mod 21) x = 7 (mod 25) b) 3y = 9 (mod 12)…

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Q: (a) Does j ≡ k(mod n) imply a^j ≡ a^k(mod n) for all a ∈ Z? Justify your answer.(b) Does a ≡ b(mod…

A: (a) Does j≡k(modn) imply aj≡ak(modn) for all a∈Z Answer: Yes, j≡k(modn) implies aj≡ak(modn) for all…

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: a) If a| (21b−c) and 6c≅3b(mod2a), then a| 41c b) If a≅b(modc) and d|c then ?≅?(mod?) c) ∴(d|…

A: this is a question of modular arithmatic.

Q: 4 If a = b (mod n), then a" = b™ (mod n) for every positive integer m.

A: It is given that  a≡b (mod n), then am≡bm (mod n) for every positive integer m. It will prove by…

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

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Q: III) Are there positive integers x and y suct that x³ = 2² +15? (Hint: for every integer x, x³ 0, 1,…

A: We shall show that no such positive integers exist.

Q: IfOcy (mod n) and e> 0 then a) *= y (mod cn) b) x= y (mod n) ) x=y (mod) d) None of these :Select…

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

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Q: Prove that ifa≡b(modn)andb≡c(modn), thena≡c(modn). Note that you must use thedivides definition of…

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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: Show that if a is an even integer then a² = 0 (mod 4) and if a is odd integer then a² = 1 (mod 4)

A: We know that an even number is a multiple of 2 and we write an even number k as k=2m for some…

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? =…

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

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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 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: 5) Which are true and which are false? a) 3XER,x^2-4x+5=0 b) VxɛZ,x^2 =0(mod4)orx^2 =1(mod4) c) Vx E…

A: In the given question we have to tell about some given statement which of them are true or which of…

Q: 5 how that Cm d n+2 Che prine 4C(n-1)! +)+n = 0 mod @ncnt2)

A: We show that the n and n+2 are primes If and only if  4((n-1)!+1)+n≡0 mod n(n+2) We take n as prime…