Problem 9. Let n ∈ Z. Show that n(n + 1)(n + 2)(n + 3)(n + 4)(n + 5) is divisible by 3, 8, and 5.

Q: Find Im(z) if z(8 + i) = 7+ (384 + 448i)

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Q: Suppose that A, B, C, and D are digits from 1 to 9 and that A + B = CD (a) What digit must C be?…

A: Given that  A, B, C and D are digits from 1 to 9 such that  A + B = CD   (a)  If 1 ≤ A, B ≤ 4 then A…

Q: Question 17 What is the expanded form of (3x +4)*? 8 Lx+256 B. 81x + 432x + 864x+768x + 256 .2 C.…

A: Option B is correct.

Q: greatest common divisor(g.c.d)

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Q: 23. If 6p-4 is divisible by 7, where p is an integer, which of the following is also divisible by 7.…

A: Option a is correct answer

Q: 4. Find the union and intersection of these families, and prove your answers. (a) A = {A, ne N},…

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Q: 1. Prove by induction that for all n € N>1 we have (a) (b) (c) 2 +3 2.3 9+ (9 × 10) + (9 × 100) ++…

A: (a) Given 2n+3<2n·3 For n=1 L.H.S=21+3=5 R.H.S=21·3=6 Thus the inequality hold for n=1 Let…

Q: (15) Let a, b = Z. Prove that if a is divisible by 4 and b is divisible by 4, then: (a) ab is…

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Q: 4. Suppose that a positive integer is written in decimal notation as n = akak-1 a2a1ao where 0 <a;…

A: Here, we use theory of congruences.

Q: 4. Prove that if n is an integer and n is a multiple of three (that is, n = 3k for some integer k),…

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Q: Question 6. We can use (C) to represent C(n, k). Show that (") is divisible by n if n is a prime…

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Q: 1 Use the factorization A = PDP' to compute A^, wherek represents an arbitrary integer. а 11(b - а)…

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Q: 7. By Principle of Mathematical Induction prove 5º – 1 is divisible by 2 for all n>1

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

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Q: 3. How many zeros at the end does the number 2022! have? (n! is a product of all positive integers…

A: Given,Number = 2022!To find: Number of zeros at the end (trailing zeros) of the given number  As per…

Q: (a) Prove that for all integer n 2 1, 3|n(n² + 11).

A: a.          In the given problem we have to prove that for all integer n≥1, 3|n(n2+11). To prove…

Q: a) Suppose that in the binomial expansion of (1 + m2)", where m and Kare non-zero constants, the…

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Q: 9. Prove that n is odd, then n² - 1 is divisible by 8.

A: In this question, we will prove the question.

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

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

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Q: (16) Prove that 22n – 1 is divisible by 3 for all n € N.

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Q: "For all integers n, n(n² – 1) is divisible by 6." (You may use quotient-remainder theorem and…

A: Let us consider, for all integers n a = n3 – n a = n (n2 – 1) a = n (n + 1)(n – 1)