Let A be the set of all positive integers whose last two digits form a number that is evenly divisible by 4. In the case of a one-digit number, we take its tens digit to be 0. Let B be the set of all positive integers that are evenly divisible by 4. Show that A = B.

Q: If x is any positive integer , then the sum of 6x and 11x is always divisible by what?

A: x is a positive integer. Find sum of 6x and 11x is divisible by what?

Q: count the number N of all sets A := {1, x2, X3, X4} of non-negative integers satisfying Xi + x2 +…

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Q: What is the set Z+ of all positive integers?

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Q: Show that for any integer n, n² – 2 is not divisible by 3.

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Q: 4) Prove that the product of any three consecutive integers is divisible by 6.

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A:

Q: Determine integers x and y such that (а) gcd(51, 288) — 51х + 288у (b) gcd(357, 629) = 357x + 629y

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Q: Let A be the set of five-digit numbers where only the digits 1, 2 and 3 are included. For example…

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Q: Let n be a positive integer. Show that in any set of n consecutive integers there is exactly one…

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Q: Prove that if x is an integer, then x2 has the same parity as x

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Q: Show that an integer N is divisible by 9 if and only if the sum of its digits is divisible by 9

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Q: Show that for any integer n , n2 – 2 is not divisible by 3.

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Q: Suppose a set of 8 numbers are selected from the set: {1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13;…

A: It is given that 8 numbers are selected from the set 1,2,3,4,5,6,7,8,9,10,11,12,13,14.

Q: Show that if m is an integer then 3 divides m3−m.

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Q: 3. List up all elements in the set A\B where A = {r € Z[0.01 < 2² < 100}, B= {r € Z]0.01 < r³ <…

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Q: Find all integers x + 3 such that x – 3 is divisible by x – 3. |

A: To find all integers such that x^3-3 is divisible by x-3

Q: For any non negative integer n , prove that an − bn is divisible by a−b,where a and b are any…

A: answer

Q: Every positive integer is the sum of at most two squares and a cube of nonnegative integers. O True…

A: consider 0  0=02+02+03 now  1 can be written as  1=02+02+13 now consider 2 2=12+02+13 now consider 3…

Q: Prove that if for some integers a, b, c we have that a° + 6° + c° is divisible by 9, thên at least…

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Q: In the 1989, Journal of Number Theory paper, N. Tzanakis and B. de Wegner showed that the largest of…

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Q: Show that for all integers m and n, if the remainder is 2 when m is divided by 4 and the remainder…

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Q: For each positive integer n, find with proof GCD(220° - 220, 220" + 218).

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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: Let A be the set of odd integers, and let B be the set of positive integers less than 10.  Find A∩B

A: our objective is to find A intersection B

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

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Q: Let X = {x;} be a set of n positive integers. Show that the sum of the integers of a subset of X is…

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Q: Let x / y= xy + x+ y be a binary operation in the set of integers. What is 4 / - 2 ?

A: Given x/y=xy+x+y  With respect to the above binary operation, we have to find 4/-2

Q: In the set R of real numbers, define C, = (n,n+1) = {xeR|n < x <n+1} for each integer n. Determine…

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Q: If the sum of two counting numbers is an even number, then the product of the two counting numbers…

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Q: The set of positive integers is ordered-isomorphic to the set of even positive integers. ylgn ihi

A: Given a statement

Q: Use a direct proof to show that the sum of two odd integers(k and l) is even.

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Q: The set of all positive multiples of v2 is countable.

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Q: Prove that it is impossible to choose three distinct integers (a – b)|(b – c), (b – c)|(c – a), (c –…

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Q: Let m and n be positive integers.

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Q: Suppose, a computer has found, for a given set of values of X1, X and X3, 12 = 0.91, 13 = 0.33 and…

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Q: Prove that for any odd positive integer, n, the sum of n consecutive positive integers is divisible…

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Q: Exercise 5. Prove that given positive integers a and b, there is a positive integer d such that • d…

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Q: a2 − b2is divisible by 5.

A: First observe that one's place of the square of a positive number can only one among 0,1,4,9,6, and…

Q: Let A be the set of five-digit numbers where only the digits 1, 2 and 3 are included. For example…

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Q: Find the positive integers n such that , the sum of positive divisor oln) = 24,

A: Sum of Divisors: Let the prime factorization of n be n=p1a1×p2a2×...×pkak Then the sum of divisors…

Q: For any non negative integer n , prove that an – bn is divisible by a-b,where a and b are any…

A: Use the binomial factorisation

Q: a) Find the number of positive integers not exceeding 120 that are divisible by 3 or 4. 1 0 01 0 0 0…

A: As per our guides lines I answered only 1st part

Q: What is the set Z of all integers?

A: The set of all integers is, ℤ=........, -4, -3, -2, -1, 0, 1, 2, 3, 4, ........