25. How many positive integersns 20000 have the properties that 2n has 64 positive divisors including 1 and 2n, and 5n has 60 positive divisors including 1 and 5n? (A) 4 (B) 5 (C) 3 (D) 2 (E) 6

Q: Compute A - 613 and (613)A, where A = - A-613 = ☐ 6 - 4 3 2 -7 - 3 1 3

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Q: Use the factorization A = PDP Ak= - 7 40 - 2 11 = 45 1 1 1 01 30 to compute A I -1 5 1 - 4 1 where k…

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Q: How many positive integers less than 10000 are there which contain at least one 7 or at least one 9…

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Q: In (5 – x) – In (5 + x) = -In 9

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Q: What is the converse of: an integer has divisors other than 1 and itself, if an integer is…

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Q: 16. Given an odd integer a, establish that a² + (a + 2)²+(a + 4)² + 1 is divisible by 12.

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Q: k+4 n(3n +7) k(k +1)(k + 2) 2(n+1)(n+2) k=1

A: Given ∑k=1nk+4k(k+1)(k+2) To show  ∑k=1nk+4k(k+1)(k+2)=n(3n+7)2(n+1)(n+2)

Q: A number n is divisible by 12 if it satisfies ∃k ∈Z(n = 12k) Give a direct proof that, if n is…

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Q: Show that the the greatest common divisor gcd(16" – 4",17" – 2") is divisible by 3 for all n E N.

A: Consider the provided question,

Q: divisible by 3 for

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Q: 9. If an+1 = 2an + 2 and ao = 4, then an = O 3(2)n + 1 6(2)n-2 O 2n + 3 O none of these

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Q: 3. We say that an integer c is a common divisor of integers a and b if cla and cb. We say c is the…

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Q: If k is a positive integer, what is the remainder when (k+ 2)(k3 - k) is divided by 6? Stilo…

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Q: (a) Suppose that there are k distinct positive integers that divide an odd positive integer n.…

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Q: 2. Prove that n(n + 1)(n + 2) is always divisible by 6

A: We have to show that the number nn+1n+2 is always divisible by 6 for all n∈ℕ. Let Pn=nn+1n+2 where…

Q: 10. Prove that n³-n is divisible by 6 for every integer n.

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Q: 10. If an+2 = 4an+1 - 3an and ao = 5, a₁ = 11, then an 4(2)n + 3n none of these O2 + 3n+1 2(2)n +…

A: In this problem we are to find the recurrence relation for an . The problem is solved step by step.…

Q: What is the value of the coefficient of the 4th term in the expansion of (a + b)¹2?

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Q: (5) (Extra credit, 5 points) Show that 102n-1 +1 is divisible by 11 (namely, 102n−1 +1 = 11 ·an,…

A: Step 1: Step 2: Step 3: Step 4: