1. Prove: 11" – 6 is divisible by 5 for all positive integers, n.

Q: 2. Prove by mathematical induCHon that १) के) 22 32 42 2n is Hrue For all integers nz 2

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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: Prove 22n! for n=4 Prove R

A: According  to guidelines  we solve one question. We solve question 1 by induction method.

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

A: To show  33n+3 - 26n - 27 is divisible by 169.

Q: Show with the help of Fermat’s little theorem that if n is a positive integer, then 42 divides n7 −…

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Q: 2. Prove thatn+ 100 > 3" if n is a positive integer with 1 <n< 3. (Hint: Proof by cases)

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Q: B) Prove that (O) = ) CE) n –k

A: We use the combination formula to simplify both sides

Q: 1. Show that there are no integers m and n such that VI1. (Hint: Prove by contradiction that we…

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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: Prove that (n – 1) +n° + (n + 1)° is divisible by 9 for n E {1,2,3, . }

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Q: 7. For n > 1, verify that 1? + 3? + 5² + ...+ (2n – 1)? = ( 2" *1 ) %3D -

A: We have to prove that for n&gt;=1 in question number (7) Sum of squares of first n natural numbers…

Q: 4. Prove that if m and n are positive integers and x is a real number, then %3D m

A: We know ⨽x⨼ = x if x is an integer. Here, since m and n are integers, we have the following steps:…

Q: 1. Prove that 5 is irrational.

A: Consider the provided expression, 5 Show that the above expression is irrational. We proof by…

Q: prove (1+ )" in <4 for all n21

A: In the given question we have to prove that  1+1nn&lt;4∀n≥1 . we can prove this by two methods one…

Q: Prove that if  n  is a positive integer, then  5 | (6^n − 1)

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Q: 5. Prove that (n, n + 2) = 1 is n is odd and (n, n + 2) = 2 is n is even.

A: Suppose (n, n+2) = d =&gt; d | n and d | n+2 =&gt; d | (n+2) - n =&gt; d | 2 =&gt; d = 1 or 2.

Q: 7. Prove: Vn,meZ, if n and m are odd, then n·m is odd.

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Q: 6.20. For which integers n > 2 does there exist a o e A, such that |o|> n?

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Q: Suppose thatk and n 2k are fixed positive integer. Justify the identity k+1 jk

A: to prove ∑j=knjk=n+1k+1above equation can be written as S2(j,k)=S2(j-1,k-1)+k.S2(j-1,k)count the…

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: Prove that 10 134n+1 + 115n+2 – 4 for all n E N.

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Q: 5. Prove that the integer 3". 22" – 1 is divisible by 11 for all n > 0.

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Q: Prove the following (by using contrapositive): Let a, b and n be integers. If n doesn’t divide ab,…

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Q: on 'N" en Prove Delween Aborms is lenre equala bub

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Q: Prove that 4.999...=5.000...=5

A: Proof:Let x = 4.999, then 10x = 49.99.

Q: 4. Prove: if an/bn (Here, as always, L represents a finite number, not oo.) L, ba #0 for any n, and…

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Q: (d) Prove by induction that 5 "– 1 is divisible by 5 for all n e Z +.

A: we have to show that 5n-1 is divisible by 5 for all n belongs to Z+ by induction for n =1.

Q: Show that for any nez, n is erther odd or even

A: Suppose P(n) is a given statement, then P(n) is true for all positive integer if: P(1) is true. If…

Q: Suppose that k and n > k are fixed positive integer. Justify the identity n n + 1 %3D k +1, j=k

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Q: 5. Use the direct proof to show that " If n is an odd integer, then n2 + 3 is even

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Q: Show that for every integer n, f (n) = (2 n3 + 3 n – 10 n + 5 n) is divisible by 7. %3D (Hint: use…

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Q: 4. Prove that the cube of any integer can be written as the difference of Notice that squ n³ = (1³ +…

A: To prove that cube of any integer can be written as the difference of two sqaures.we have given hint…

Q: prove by contrapositive : for all n,m integers if nm is even then n is even or m is even

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Q: Q. Prove that by mathematically Induction 2" < (n + 1)! , for all integers n 2 2. а.

A: We Solve it by First Principle of Mathematical Induction.

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: 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: * 10. Is the following proposition true or false? Justify your conclusion. For each integer n, n is…

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Q: Suppose n E Z. Prove that if n is even, then n² + 7n + 10 is even. (Use Direct Proof)

A: Suppose that n∈Z and let n is even.To show: n2+7n+10 is also even.

Q: 1. Prove that if 7n2 + 5 is even, then n is odd

A: The given statement is,

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: 15. Prove the following identity C) + (" ; ') + (" * ) + (" ; ) +-+ " : )--- " * ** ') (n + k + k…

A: Binomial Coefficient. The binomial coefficient is the number of ways of picking unordered outcomes…

Q: 4. Prove that if m and n are positive integers and x is a real number, then

A: We have to prove here that the expression here follows both commutative and associative property. If…

Q: 4. Prove that given any integer for n, n3 + 2n will be divisible by 3.

A: We have to prove by induction.

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

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

A: We show that 5m-1|5n-1 whenever m|n ,for any positive integer m,n

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: 4. Prove by mathematical induction that = 2+1 1. 20+2¹+2²+...+2"

A: We have to find solutions

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

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Q: 4. Use a proof by contrapositive to prove that, for integers r and y: if a+y is odd, then exactly…

A: Here we use  Odd + Odd = Even Even +Even =Odd   Note:According to our guidelines we can solve only…