Prove that If a and y are integers and r + y is odd , then a – y is odd. I Show that for any integer n , n² + 2 is not divisible by 4.

Q: The representation of an even integer is 2k; and that for odd is 2k+1 or 2k-1 for some integer k.…

A: To show the representation of an even integers is 2k and that for odd integer is 2k+1 or 2k-1 by…

Q: Let f(sub-n) denote the nth Fibonacci number. Show that f(sub-n) is even if and only if n is…

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Q: 2. 6n - 1 is divisible by 5 for n any positive integer

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Q: Prove 3 | 17k+1-2*5k+3+11k+3 for k>=0 (nonnegative integer)

A: Here in this problem we have to prove that for k≥0 i.e k is a non negative integer…

Q: The representation of an even integer is 2k; and that for odd is 2k+1 or 2k-1 for some integer k.

A: integer  which have 2 as one of their factors are called even numbers. The integer  which do not…

Q: 8. Find the number zeroes with which the decimal representation of 6000! terminates. (a) 1472 (b)…

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Q: Assume an unlimited supply of coins worth 5 cents and 9 cents. Prove that some

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Q: The representation of an even integer is 2k; and that for odd is 2k+1 or 2k-1 for some integer k.…

A: Even number: we say a number 'n' is even if it is divisible by 2. if  'n' is even then it is…

Q: Use Wilson's Theorem to verify that 4(29!) + 5! is divisible by 31.

A: The given number is 4 (29)! + 5! . Our aim is to verify that the given number is divisible by 31…

Q: Find a recursive definition for the set of all integers which are greater than or equal to 15.

A: Sequence is defined as an arranged element that satisfies some definition. Sequences are many types…

Q: Find the smallest positive integer in the given set. (a) {6u + 15v : u, v ∈Z}    (b) {12r + 17s : r,…

A: Given: (a) { 6u+15v:u,v∈Z }(b) { 12r+17s:r,s∈Z }

Q: Maximise Z = 5x1 + 6x2 Subject to X1 + x2 0 and integer

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Q: How many divisors does (1000000! + 1) have that are less than 1000000?

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Q: Show that the the greatest common divisor gcd(16" – 4",17" – 2") is divisible by 3 for all n E N.

A: Consider the provided question,

Q: 4. >2 1+ 100 1-0,9* >0,8

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Q: Solve for X, show all calculation in binary number system only X = ((1110)₂-(0011)₂) (10011010)₂)…

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Q: Show that 3·4n+ 51 is divisible by 3 and 9 for all positive integersn

A: Consider the provided expression,

Q: Jse generating functions to show that every positive integer can be written in exactly one way as a…

A: A generating function is a way to encode an infinite sequence of numbers by treating them as the…

Q: S(x + Vx)dx 73 6.

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Q: prove the difference of the squares of two z integers is always divisible by 16 (i.e. z1^2 - z2^2 =…

A: We prove that (z_1)^2-(z_2)^2=16k for some integer k, then we say that divisible by 16.

Q: 3 Show thet x+3x+2 has four in 76 zenos.

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Q: b using recursive Computing the greatest common divisor of two nonnegative integers a and b with a <…

A: To write the program  (R)

Q: 2. Prove that n(n + 1)(n + 2) is always divisible by 6

A: We have to show that the number nn+1n+2 is always divisible by 6 for all n∈ℕ. Let Pn=nn+1n+2 where…

Q: Find 2's complement of 001011011001100

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Q: 5 =5 Hence, 6"-1 is divisible by 5 is (10.). for n=1 Part 2: Assume that 6" - 1 is divisible by 5…

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Q: True/False: Every integer greater than 1 is divisible by a prime number. True False

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Q: Give a recursive definition for the set of all the positive integers that are less than 100 and not…

A: Given: Set of all positive integers that are less than 100. To find: Recursive definition for the…

Q: how: For all integers and , if n-m is even then  n^3-m^3  is even

A: We have given that for all integers n,m (n-m) is even.

Q: The representation of an even integer is 2k; and that for odd is 2k+1 or 2k-1 for some integer k.…

A: Given the statements: The representation of an even integer is 2k for some integer k. The…

Q: Prove {12a + 4b : a and b are integers} = {4c : c is an integer}

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Q: How many zero divisors are in Z,, where n = 32547?

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Q: How many divisors does (1000000! + 1) have that are less than 1000000? Use Wilson's theorem

A: Given: 1000000 !+1To find: The number of divisors of 1000000 !+1 that are less than 1000000 using…