Show that for all integers m and n, if the remainder is 2 when m is divided by 4 and the remainder is 3 when n is divided by 6, then mn is divisible by 6.

Q: Let x and y be positive integers such that the prime factorization of x has exactly four 5’s and…

A: We have to find how many 5’s does the prime factorization of x + y contain.

Q: Prove that for any positive integer n, 6(7") – 2(3") is divisible by 4.

A: We use binomial expansion formula for an-bn to solve the above problem.

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

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Q: the fourth power of an integer is x, and x is between 100 and 600. What is x?

A: It is given that the fourth power of an integer is x.That is, x = n^4, where n is a positive…

Q: Suppose a, b, and c are integers and suppose a | b and a{c, where a}c denotes that c is not…

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Q: R.1. Show that among 14 distinct integers, there must be two that have the same remainder when…

A: In order to prove this statement pigeonhole principle has to be applied. According to pigeonhole…

Q: Prove that for all integers a, b, and c if a divides b - 1 and a divides c - 1 then a divides bc…

A: If a number x divides y then y is a multiple of x.

Q: If a, b, and c are integers such that b is a multiple of a³ and c is a multiple of b², then c is a…

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Q: show that n(n^2-1)(n+2) is divisible by 4 for all integers n

A: To show that n(n^2-1)(n+2) is divisible by 4 for all integers n

Q: Show that if n is any integer, then 3 divides n3 − n

A: To prove:  if n is any integer, then 3 divides n3 − n. Concept used: Here we shall use the identity:…

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

A: Let a=anan-1...a3a2a1 be an integer and S be the sum of the digits. Consider a,…

Q: a

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Q: Prove that for a positive integer n we have that 33n+3 – 26n – 27 is divisible by 169.

A: Topic - principle of mathematical induction

Q: If p + 2 is a prime number, prove that p² + 3 is divisible by 4.

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Q: Prove that for any integer n, we have n - 2 is not divisible by 4.

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

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Q: does there exists   ______such that if n is divided by 5 the remainder is 2 and  if   ?

A: We have to check whether there exist n such that if n is divided by 5 then remainder is 2. Let if n…

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

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Q: If m times the nth term of an AP is equal to n times its nth term, show that the (m + n) th term of…

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Q: Prove that if for some integers a, b, c we have that a + b3 + c* is divisible by 9, then at least…

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Q: Show that for any integer n > 5, the integers n, n+2, and n +4 cannot all be primes.

A: Lets prove by contradiction Lets say integers n, n+2 and n+4 are all prime number for some integer…

Q: Prove by mathematical induction that for all integers n 2 0: a 5" + 2 x 11" is divisible by 3, b 33n…

A: (a) Case 1 : n=1 51+2×111 = 5+2×11= 5+22= 27 As 27 is a multiple of 3 , hence the result holds for…

Q: Suppose a and b are integers that divide the integer c. If a and b arerelatively prime, show that ab…

A: Since a/c   &  b/c  , we know there exists integers K1 & K2  ,with K1 a=K2 b=c--1Also…

Q: If a, b,n are integers and a^2 + b^2 = n, then the remainder of n upon division by 4 is not 3.

A: Here a and b are two integer then n is the sum of two square. There is only one case n to be prime…

Q: Suppose that all the entries in A are integers and det A = 1. Explain why all the entries in A–1 are…

A: Inverse of a matrix A is calculated by,

Q: For any positive integers m, n (

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Q: that if 3 divides the sum of the digits of n, then 3|n

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Q: If n = 4k + 3, does 8 divide n2 - 1? (Assume that n and k represent integers.) O Yes, because n2 - 1…

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Q: n + 5 > 4 n

A: We know, the square of an integer is always non-negative (n - 2)² ≥ 0 for all integers n

Q: Let n = integer; 7 divides n if and only if 7 dividesn^2 (The result(s) ought to be integer(s))

A: Consider an integer n. We have to prove that 7 divides n if and only if 7 divides n2. Suppose that 7…

Q: Prove that n2+n  is  divisible by 2, where nN.

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Q: If n is a perfect number, prove Σ d|n 1 2.

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Q: Find all integers n> 1 such that 1" + 2"+... +(n – 1)" is divisible by n.

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Q: "For all integers n, n(n² – 1) is divisible by 6." (You may use quotient-remainder theorem and…

A: Let us consider, for all integers n a = n3 – n a = n (n2 – 1) a = n (n + 1)(n – 1)

Q: Show that 3-4" + 51 is divisible by 3 and 9 for all positive integers n.

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Q: Prove that 18 divides 2. 192n+1 16 . 174n+1 for all n e N.

A: We need to show that 18 divides 2·192n+1-16·174n+1 Now, a divides b is there exists an integer t…

Q: Find all positive integers n for which n1º +1 is divisible by 10.

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Q: (a) Prove that 82n – 12" is divisible by 13 for all nɛ N.

A: As per guidelines we are supposed to solve maximum one questions at a time.kindly post another…

Q: If  n×k divisible by 4, then n is divisible by 4 or k is divisible by 4. Which of the following is…

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Q: Show that for any integer n ≥ 2, C(2n,n) > C(2n,n-1)

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Q: Suppose that p is a prime number greater than 30. Prove that when p is divided by 30, the remainder…

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Q: Prove that for a positive integer n we have that Bn+3 - 26n – 27 is divisible by 169.

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Q: Suppose a, b, c and d are integers. Prove that  if a divides b and c divides d, then ac divides bd

A: According to the given information:   It is given that a divides b, there is an integer x such that…

Q: Let suppose integers are randomly selected. Prove that the difference of two of them is divisible by…

A: Solution

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: 4. Suppose that a and b are integers that divide the integer c. If a and b are relatively prime,…

A: Suppose that a and b are integers that divide the integer 'i'If a and b are relatively prime. Show…

Q: divisible

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