Al. a. Find an integer that leaves a remainder of 2 when divided by either 3 or 5, but that is divisible by 4?

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: Prove that N × N × N is countable. Does your argument generalize to the Cartesian product of k…

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

A: Let p≠2 be a prime number.

Q: An integer n > 2 is composite if and only if n doe

A: Given Statement: An integer n > 2 is composite if and only if n does not divide (n - 1)! + 1.…

Q: (a) Show that n(n2 – 1)(n + 2) is divisible by 4 for all integer n.

A: As per guidelines , if there are more than one questions , we have to solve the first question only…

Q: 1. Compute the value of (), for n > 3.

A: Formula: nr=n!r!n-r!

Q: Let p > 5 be the prime number. Prove that the expression p – 10p +9 is divisible by 1920.

A: We have to prove that the expression p4-10p2+9 is divisible by 1920.

Q: Suppose that [E:Q] = 2. Show that there is an integer d such that E = Q(square root of d) where d is…

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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: 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: Use ibn al-Haytham's procedure to derive the formula for the sum of the fifth powers of the…

A: We have to derive the formula 15+25+......+n5=n66+n52+5n412−n212 by using ibn al-Haytham's method…

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: Let n ∈ ℕ. Show that either √n is either an integer or it is irrational.

A: We have, n∈N Assume that  n=a2b2,where a,b are positive integers with no common factors other than…

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

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Q: Suppose a is an integer. Prove that if a^3 is a multiple of 3, then a is a multiple of 3.

A: Let a3 is a multiple of 3 but a is not a multiple of 3.a3 is a multiple of 3Therfore, 3 divides…

Q: For which prime p,  p^2+2 is also prime?

A: Put p=3 in p2+2 and obtain that, 32+2=9+2=11. Clearly, 11 is a prime number. Thus, p=3 satisfies…

Q: The small positive integer n for which (1+i)?"=(1-i)?n is =..

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Q: Show that the cube for any integer is of the form 7k, or 7k ± 1

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Q: If n is a natural number, the number 34n+4 – 43n+3 is divisible by: -

A: Given : To find : A positive number which can divide above expression

Q: A. If p 2q25 is a prime number, prove that p-q is divisible by 3.

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Q: *) The following number has a missing digit x, n = 13645x04142250 a) If n is divisible by 9, then x…

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Q: If p is a prime number greater than 3, which of the following represents the sum of the positive…

A: we will check all the options to ensure which is correct.

Q: If z ≥ 2 is an integer, show that the following numbers are composite. n! + 2, n! + 3, n! + 4, ... ,…

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Q: If a is any integer, is a(a + 1) even or odd?

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Q: If n = p,P2 ... P where the p, are distinct prime numbers, how many ways can n be written as a…

A: First let us formulate the problem in a different way. Look! we have to find the total number of…

Q: Prove or disprove the following statement: the product of any 4 consecutive integers is divisible by…

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Q: For any integer a, show that a²-a+7 ends in one of the digits 3, 7, or 9.

A: For any integer a , we have to show that a2 - a + 7 ends in one of the digits 3 , 7 , or 9 . Note…

Q: 10. If a, b and p are positive integers, what is the LCM of ap, bp^3 and a^2p^2? 1 a^2bp^3 abp p^2

A: Since the student has mentioned to solve multiple questions (no. 10-12), but we can solve one…

Q: (b) An integer is divisible by 3 if and only if the sum of its digits is divisible by 3.

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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: Prove or disprove: If an integer is divisible by 6 and by 10, then it is divisible by 60.

A: Answer

Q: For any positive integers m, n (

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Q: Suppose “a” is an integer. If 5|2a, then 5|a.

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

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Q: If m is a positive integer, does 6 divide (2m)(m+2)(3m+3)? True False

A: Here, the given number is: (2m)(m+2)(3m+3)=2(m(m+2))3(m+1)=6m(m+1)(m+2) because multiplication is…

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

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Q: Prove the given statement: If the product of two integers is not divisible by an integer n, then…

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Q: R.2. Show that among 29 distinct integers, there must be a set of 5 integers all of which have the…

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Q: Let m and n be positive integers.

A: Given, m and n be positive integers.  To prove that mℤ⊕nℤ is an ideal of ℤ⊕ℤ and that…

Q: 11. Prove that for each integer a, if a? - 1 is even, then 4 divides a?

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Q: C) Prove that the product of k consecutive positive integers is divisible by k!

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Q: Use trial-and-error to determine the smallest integer n for which the given statement is true. 6n <…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Let p > 5 be the prime number. Prove that the expression p* - 10p? + 9 is divisible by 1920.

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Q: Is it true that for any odd n the number 2" +1 is divisible by 3? Prove your answer.

A: We prove this by using PMI(Principle of Mathematical induction)

Q: Prove that if an integer m has the same parity as an integer n, 5m + 7n is even.

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

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Q: For all n>= 1, then P(n) = n-n is divisible by 3 Select one: O True O False

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Q: ove that n’ +5n is divisible by 6 for all n E N.

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