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

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: For any positive integer n, n(n+1)(2n+ 1) EP = 12 +22 + .. + r?

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Q: if 3 does not divide n then 3 divides 2n^2+1

A: Let us Assumue 3 divides (2n2+1) n2+1=3λ Now  2n2+1=n2+1+1⇒(2n2+1)+(-λ)3=1⇒ gcd(2n2+1,3)=1…

Q: If n is an even postive integer, then 3n+2^n-2 is odd.

A: Number problem....

Q: Find a positive integer n for which the equality holds = 1 2

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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: For positive integers m > n , show that -2 + n- 1 -3 + ...+ n + n-1 n - 1 + in

A: The right-side of the given equation can be expressed as…

Q: Let a and b be integers. Show that if every integer that divides a also divides b, then a | b. (a…

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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: If n is an integer, then 2n + 1 is odd

A: Given , n is an integer We have to show that 2n+ 1 is odd

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: If n is positive integer, Find (v3 + i)" + (V3 – i)".

A: To find the value of: 3+in+3-in Note that z=x+iy can be written as: z=r eiθ, r=x2+y2, θ=tan-1yx And,…

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: Prove that if n is a positive integer, then 133 divides 11"+1 + 122n-1

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Q: For all integers n, if n2 is odd then n is odd.

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

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

A: To find all the integers n>1 such that 1n+2n+...+n-1n is divisible by n

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: Find all integers n such that n| (2n+3). *

A: Given, n|(2n+3)     TO find, All integers n

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: Show that if n is a positive integer then =n(n + 1)2"-2

A: We know the binomial theorem formula isx+yn=∑k=0nCkn xn-kyk, where Ckn is the binomial co-efficient…

Q: For each positive integer n 2 0, let P(n) be the formula 3n+1 – 1 1+3 + 32 + ... 3n 2

A: We prove this formulas by using PMI

Q: What is the smallest integer n for which 25n>512?

A: The provided expression is as follow:

Q: For how many positive integers 'n', n3-8n2 + 20n - 13 is a prime?

A: Introduction :

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: For every integer n > 0, 1+2+22 + · + 2" = 2"+1 – 1.

A: Given: The statement 1+2+22+⋯+2n=2n+1-1 Proof: Base step: Show the statement is true when n=0.…

Q: se indirect proof to show that, "If n is an integer and 3n + 2 is even, then n is even

A: Let us solve the classical algebra problem in the next step.

Q: For every integer n > 1, 2+4+6+ + 2n = n² + n.

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

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Q: that for any integer n, n -n is even.

A: Prove that for any integer n, n3-n is even.

Q: Show that if n is a positive integer, then (n + 1, n2 - n + 1) = 1 or 3

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Q: Prove that the statements here are true for every positive integer n. a + (a + d) + (a + 2d) + · · ·…

A: To Determine: Prove that the statements here are true for every positive integer n. Given: we have…

Q: Show by contradiction that for any integer n: if n>0, then n+10 n+5

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

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Q: How many integer solutions to x1+ x2+ x3+ x4 = 36 are there for which æ1 > 5, x2 > 3, x3 > 5 and æ4…

A: We have to find the number of solutions to x1+x2+x3+x4=36 for which x1≥5, x2≥3, x3≥5 & x4≥6.

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

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

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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: Show that 5 | n5-n for all integers n.

A: To prove the above statement we will use the principle of mathematical induction which is divided in…

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

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Q: If n is an even integer, then 3n° is an even integer.

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Q: Show for all integers n2 0 that 91 (4" - 1). a.

A: A. We will solve the question using the method of mathematical induction.

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

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Q: 2 ) + ("**) = n²- .2 2

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Q: (c) For each integer n, if n is odd, then 8 divides (n4 + 4n² + 11).

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Q: if n zo is an integer, then n!=n(n-)3.21 ...

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Q: 5. Find all integers 100 ≤ n ≤ 999 such that the last three digits of n² is same as n.

A: To find: All integers  100≤n≤999 such that the last three digits of n2 is same as n.