Let m, n be positive integers. If m n, prove that 5m - 1 5" - 1

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

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Q: Let m, n >0 be integers. Prove that if m +n 2 29 then m > 15 Question 11 or n> 15 by contradiction.

A: To prove m+n≥29 then m≥15 or n≥15 where m,n≥0 be integers.

Q: Prove the following statement. 1) if k is a positive integer, then 3k+2 and k+1 are relatively…

A: We have to prove: If k is a positive integer, then 3k+2 and k+1 are relatively prime.

Q: Use direct proof to show, "If m + n and n + p are even integers (s and t), where m, n, and pare…

A: To answer the given questions we shall use the following definition  An integer n is said to be even…

Q: 11. Prove that for each integer a, if a² – 1 is even, then 4 divides a² – 1.

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Q: 2. (a) Find integers a, m, n such that m | a and n | a, but mn /a.

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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: 23. Prove that for any integer a, 9/ (a² – 3). -

A: Prove that for any integer a, 9 ⫮ (a2 – 3).

Q: Show that, for given integers a, b, c, if gcd(a, c) = gcd (b, c) = 1, then gcd (ab, c) = 1

A: Given a,b,c∈Z+ And gcda,c=1 and gcdb,c=1 We have to show that gcdab,c=1 Since, gcda,c=1 and gcdb,c=1…

Q: (4) Prove in two different ways that if a is a positive r number, then (a + 1)" > a" + na"-1:

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Q: Let a, b be two positive integers with (a, b) = 1. If ab = c, then prove that there exists

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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: 2. Ifr + 1, show that for any positive integer n, a + ar + ar +...+ ar" = a(rn+1 – 1) - r - 1 the…

A: we need to prove if r≠1 thena+ar+ar2+.....+arn=arn+1-1r-1let pn=a+ar+ar2+.....+arnto prove…

Q: show

A: Given : m+n and n+p are odd  So, we can write, m+n = 2s+1 and, n+p = 2t+1, where, s & t are…

Q: Show that if a1,: a2., an are integers that are not all 0 and c is a positive integers, then…

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Q: 9. Find a pair of positive integers a<b such that a+b=557685 if possible. Ans:

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Q: Let m, n are integers. Prove that if m*n is even, then m is even or n is even. Specify what type of…

A: Given: m,n are integers. To prove: If m*n is even then m or n is even.

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: Show that if a|m and a|(ms+nt) for some integers a≠0, m, s, n, and t, then a|nt.

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Q: Prove that for all the integers a and n, if a | n, then a^2 | n^2

A: Given that a | n. This means n=ak, where k=integer. Then we square both sides: n2=(ak)2 or n2=a2k2

Q: 11. Prove that if d divides a, then dk divides ak for every positive integer k.

A: What is Divisibility: Divisibility is the base of number theory. For two given integer with one…

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

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Q: 3. Show that for all integers n: a) if d is an integer such that d |n + 8 and d | n² + 1, then d |…

A: To solve the given questions we will use the following formulas : 1.   We say that two positive…

Q: Establish that if a is an odd integer, then 2" а = 1 (mod 2" +2) for any n 2 1.

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Q: Prove that 36n is not O(35n)

A: Let, 36n∈O35n

Q: Show that if a and b are positive integers, then ab = lcm(a, b) . gcd(a, b).

A: Consider two cases: 1. a and b are multiples of n i.e. they have a common factor n So a=nx, b=ny…

Q: m and n are relatively prime if and only if am+bn=1 for some integers a and b.

A: It is given that :am+bn=1--(1) for a,b,m,n ∈ Z Now, if  k divides both m and n, then there exist p…

Q: Let m, n E Z. Prove that if m and n are odd, then m + mn +n is odd. Use the definition of even/odd…

A: If a number n is odd then  n=2k+1 for some k in integer.

Q: Let m and n be positive integers and let k be the least common multipleof m and n. Show that mZ∩ nZ…

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Q: Give a direct proof of the claim: V n E N, Vk E N, if the last k digits ofn are divisible by 2k then…

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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: Suppose that m and n are relatively prime and r is any integer. Showthat there are integers x and y…

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Q: Prove that 3|(52n – 1) for all ne N.

A: We prove it by mathematical induction. 1st step We show that the statement is true for n = 1.…

Q: If m and k are positive integers, then, then Ø(m^k )= *

A: The objective is to state whether the statement is true or false. a) The statement: The quadratic…

Q: Let n and m be integers. Prove that if m + n >= 100, then either m >= 50, or n >= 50.

A: Prove the statement by proving the contrapositive. The contrapositive of the given statement is as…

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: 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: Prove that if m and n are even integers, then mn+ 1-m-n is an odd integer.

A: Given :- m and n are two even integers. To prove :- mn + 1 -m -n is odd integer

Q: Let m >1 be an integer, and let R = Z/mZ. Prove that R is nice if and only m is prime. Proof. ...

A: Given: m&gt;1 is an integer and R=Z/mZ. To show R is nice if and only m is prime.

Q: 1. Prove that if m + n andn +p are even integers, where m, n, and p are integers, then m + p is…

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

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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.

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Q: Establish that if a is an odd integer, then 2" = 1 (mod 2" +2) for any n 2 1.

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Q: Prove that if m and n are integers and mn is even, then m is even or n is even.

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Q: 4. Let bo = 1, bị = 3, and for each integer k > 2, we have bk = 4bk–1 – 4bk–2 Prove that for all…

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Q: Prove that if p,q, and r are integers, r|p and r|q, then r^2|pq.

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