Let m be fixed positive integer. Then an integers ais congruent to an integer b modulo m, if: a-blm o mla-b o m|b-a o both mla-b and mlb-a

Q: 1) Find an integer a and an integer y, such that x = 5 mod 10 y = 7 mod 12 x = 9 mod 21 y = 2 mod 17…

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

A: Given that m is a fixed positive integer. a and b are integers. Also, given that integer a is…

Q: Let a be and integer and b be a nonzero integer. Prove that alb if and only if Mb C Ma.

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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…

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

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

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Q: Let a, b, e, m, and n be integers such that aJb and a l e. Prove that a I (mb + ne).

A: Let a, b, e, m, and n be integers such that ab and ae. The objective is to prove that amb+ne. We…

Q: show that for eury A s integer nz3; A n- Am-2 +A2J. Mence determine ASO.

A: To show: That An=An-2+A2-I for n≥3 then find A50. Given: The matrix is A=100101010. Calculation:…

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

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Q: a be an odd integer. Pro

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Q: (a) Let n be any integer. Prove that n2 is congruent to 0, 1 or 4 modulo 8.

A: Note: According to bartleby guidelines we have to answer only first question please upload the…

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

A: Definition: Let a, b∈Z  and m be a positive integer. then we say a is congruent to b modulo m if and…

Q: Suppose that a is an odd integer and (a, 91) = 1. Prove that a12 ≡ 1 (mod 1456).

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Q: Prove that ifn > 1 and a > 0 are integers, and d = GCD(a,n), then the additive order of a modulo m,…

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Q: Prove that if p is a prime and a is any integer so that p does not divide a, then the additive order…

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Q: Show that if a, b, c, and d are integers with a and c nonzero, such that a|b and c|d, then ac|bd.

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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: Let a, b be integers for which a01 and ba02 are relatively prime. Show that a and b are also…

A: Given: a and b are two integers for which a101 and b202 are relatively prime. We have to show that a…

Q: 5. Let n be a positive integer, and assume that a and b are integers such that a = b (mod n). Prove…

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Q: Show that if a,b, and n are integers such that n≥2 and a≡b (mod n), then gcd(a, n) = gcd(b, n).

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Q: Supposed a and b are relatively prime integers and c is an integer such that a | c and b | c. Prove…

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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: 4. Exercise $2.5 #58a. Let a and b be integers not divisible by the prime mumber p. (a) If aP = mod…

A: Given the data

Q: ac bc (mod mc). 14. Show that if a, b, and c are integers with c>0 such that a = b(mod c), then (a,…

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Q: Suppose that m is a positive integer. Use mathematical induction to prove that if a and b are…

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Q: 3. Show that if a and b are integers such that alb, then ak|b* for every positive integer k.

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Q: Let a and b be positive integers with (a, b) = 1. Show that if g is a primitive root (mod ab), then…

A: We have given (a,b)=1 where a, b both are positive integer. We have to show if g is primitive root…

Q: Let x be an integer. Prove that, if xmod15 = 4, then 15|(x2-1)

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Q: Let n = 2p for some prime number p > 2. Show that if a is a positive integer such that gcd(n , 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 a and b be integers and n a positive integer. Assume also that a and n have a common divisor d…

A: Use the following concepts, to prove the required result. If a divides b then b is a multiple of a.…

Q: Let m be an integer with m > 1. Note that a = b (mod m) if and only if m divides a – b. Define the…

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Q: Let p = p1 ... Ps be an integer with digits p1,..., Ps. Show that 0.p1 . . · Ps %3D 10° – 1

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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: Show that if a, b, and m are integers such that m > 2 and a = b (mod m), then ged(a, m) = gcd(b, m).

A: This question is related to Number Theory

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: (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 nonnegative integer a less than 28 represented by each of these pairs, where each pair…

A: According to the given information, it is required to find the non-negative integer a less than 28…

Q: Use the definition of congruence to prove that if j, k, 1, m, nE Z are such that k =1 (mod n) and…

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Q: Let a be an integer. Prove the following statements: (a) If a is even, then a² = 0 (mod 4). (b) If a…

A: Congruence: If a and b are integers then a ≡b mod n for some n &gt;0 if and only if n| b - a.

Q: (f) If a = b (mod n), then a* bk (mod n) for any positive integer k.

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Q: (c) Use Fermat little theorem to show that (a + b)® = a® + ® mod prime p and any positive integer a.…

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Q: prove For all integers a, b, c, and d, if a|c and b|d then ab|cd.

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Q: Let x be an integer. a) Prove that, if xmod15 = 7, then 3|(x2-1). b) Prove that, if xmod15 = 7,…

A: See the detailed solution below.

Q: 2. Prove that if a, b, c, d, and m are integers such that m > 0, a = b mod m and c= d mod m, then…

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Q: Let m, n be positive integers. If m n, prove that 5m - 1 5" - 1

A: We show that 5m-1|5n-1 whenever m|n ,for any positive integer m,n

Q: If a and b are integers and n is a positive integer, prove that a mod n =b mod n if and only if n…

A: If a and b are integers and n is a positive integer,    Formula used: a≡b mod n if and only if  a-b…

Q: 4.18. Let m, ń EN such that m|n. Prove that if a and b are integers such that a = b (mod n), then a…

A: The detailed solution is as follows below:

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.