Let m be fixed positive integer. Then an integers a is congruent to an integer b modulo m, if: ab|m m/a-b m)b-a both mla-b and m|b-a

Q: m and n be positive integers so that m n. Show for any integer a that the congruence

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…

A:

Q: Let a,b be integers, not both 0. Show { na + mb : n,m ​​​​​are integers} = iff a and b are…

A: We have to prove the given statement.

Q: Let m be a positive integer. Show that a mod m = b mod m if a ≡ b (mod m)

A:

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.

A:

Q: Express each nonnegative integer a less than 15 as a pair (a mod 3, a mod 5).

A: The values of a could be, 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14.

Q: let b be a nonzero integer. prove that gcd (0,b) = |b|.

A: We need to prove that for a nonzero integer b,

Q: Let a and b be integers such that b measures a. (a) Prove that –b measures a.

A: Given: b measures a To proof: -b measures a

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: Let a, b, c be positive integers and let d = (a, b). Since d|a, there is an integer h with a = dh.…

A:

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…

A:

Q: a be an odd integer. Pro

A:

Q: |Let a, b, and c be integers. Prove that if gcd(a, b) | bc, then a | 1 and gcd(a, c) 1 and a (a) C.…

A: To prove the required divisibility properties under the given conditions on the greatest common…

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

A:

Q: •Prove: For all integers a, b, and c, if a|b and alc, then a|(b+ c). we can assume a #0.)

A:

Q: Let a be an algebraic integer in Q(v-37) and let A = (2, 1 + v-37). Prove that either a or a –- 1 is…

A: Given that,         α is an integer in Q-37 lets consider  α=-37⇒α-1=-37-1

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.

A:

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: Prove: For all integers a and b, if a + b is odd then exactly one of the integers, a or b, is odd.

A: (1) We have to prove that a+b is odd then exactly one of the integer a or b is odd. Suppose that a…

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).

A:

Q: Prove: If a, b and c≠0 are integers, then a|b if and only if ac|bc.

A:

Q: Let a and b be two integers. Prove that if a+b is even then a-b is even.

A: Given that a and b are any two integers. We are trying to prove "if a+b is even then a-b is even."

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,…

A:

Q: Let i be a positive integer and A, = [-i+1,1]. Then U2, A

A: We have to use properties of union in this question.

Q: If all entries of A and A -1 are integers, prove that <let A = 1 or -1. Hint: What is det A times…

A:

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

A:

Q: Let a, b be nonzero integers with (a, b) = 1. Compute (a + b, a – b).

A:

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

A: for a is congruent to b modulo m if n divides a-b. for eg 16 is congruent to 4 modulo 6 because here…

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

A:

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

A:

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

A:

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

A:

Q: Let a, b, c be integers. Prove that if a|b and a|c, then a|(b +c).

A: Suppose a|b and a|c. Then there are integers m and n such thatb = am and c = an.Then b + c = am + an…

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

A:

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: Prove: If a is a positive even integer and b is an odd integer, then (a,b)=(a/2, b).

A: Let d be gcd of a and b. Now, b is odd so d cannot be even or is odd.  Now, since d is odd and…

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

A:

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 >0 if and only if n| b - a.

Q: Let a, b, c be non-zero integers. Prove that gcd(a, bc) = 1 if and only if gcd(a, b) = 1 and gcd(a,…

A:

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

A:

Q: Let m, n, p be integers. Use direct proof to show that if m+n and n+p are odd then m+p is even.

A:

Q: Let a, b, and c be integers. Prove that if a divides b and a divides c, then a divides (b-c).

A: a, b and c are integers, a divides b and a divides c. Since a divides b, therefore b=ak,  k is any…

Q: If a and b are integers such that a? = b² (mod p), where p is prime integer, then a) a = b (mod p)…

A:

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

Statements and Logic

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