Problem B Show that 3. 4" + 51 is divisible by 3 and 9 for all positive integers п.

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Q: Problem 13. Decide if it is possible to have two irrational numbers, a and b, such that a is…

A: 13. Take a=66, b=6 . So, a, b both are irrational numbers.   But, ab=666=66=216  which is rational.

Q: Problem 6.4 Suppose that n is a positive integer. Justify the identity 2 2n ()

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Q: Question 5 Find all integers n >1 such that 1" + 27+...+(n – 1)" is divisible by n. |

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Q: Problem 3. Let p > 3 be prime and let 1 < qi < q2 < • · ·< qk <p -1 be the list of all quadratic…

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Q: Problem 7 Let p and q be distinct primes and let n = p•q. Show that (Z/nZ)* has an element of order…

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Q: Problem #17: 3 Carl says that 8 3>2 and 8>3 How would you help Carl? because

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Q: Problem 3: Prove (directly from the definition) that for any odd integers a and b, a? +3b is even

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Q: QUESTION 9 Use the contrapositive method to show that for any integers a and b, a+b≥13 implies that…

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Q: Problem 3. Let a be an integer that is coprime with 240. Prove that 240|a4 – 1. -

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Q: QUESTION 12 Show by contradiction that if 5x2+ 3 is irrational, then x is

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Q: Problem 3: Let B, be the number of compositions of n in which the first part is odd. If n > 1, show…

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

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A: Check for k=1 and k=2

Q: Problem 4. Let n be an integer. Show that if 2+ 2v1 + 12n² is an integer, then it is the square of…

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Q: Problem 2. Prove that if integers a and b are divisible by an integer c, c+0, then a + is divisible…

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Q: PROBLEM 2 Fnd the nuLmber of 4 letter words using the letters of the word MISTER in which atieast…

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Q: Problem 97 Let A be any set of 51 distinct integers chosen from 1, 2, 3,., 100. Prove that there…

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Q: Problem 3.3. Prove by contradiction that there exists no n e Z such that both n and n +9 are even.…

A: You have written 2n=n+9, which cannot be done. Please see the proof below.

Q: Which of the following is an Integer ?

A: Given,           Ingeger number.

Q: Problem 3. Let a be an integer that is coprime with 240. Prove that 240|a4 – 1.

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Q: Problem 4. Let n be an integer. Show that if 2+ 2/1+ 12n² is an integer, then it is the square of an…

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Q: Problem 1. Does there exist a natural number n such that 13 (n² + 1)? Prove your assertion.

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Q: How many integer solutions are there to 1 +X2+X3+X4+Xs=31 with

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Q: Problem 4. Let n be an integer. Show that if 2 + 2/1+ 12n² is an integer, then it is the square of…

A: