13. Floor and ceiling notation were defined in Section 1.1. (a) Show that []+[=n for all positive integers 2n n 21. (b) Show that []+[] = m for all integers m.

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Q: 3. Find integers x, y € [0,51] such that (a) 23x= 1(mod52) (b) 23y = 4(mod52)

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Q: n 2n + 2 /2 n 2n k k=1 \k –1, п+1 n -

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Q: 3 8.) What is the quotient of -3- and 0.1? Show all work R 10

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Q: Which of the following statement is true in ZZ? O O O (0=1+x)^AXE (0=A+X)^EXA (0=1+x)^AXA None of…

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Q: 10. Give a combinatorial proof: For n ≥ 1 and k ≥ 1, (n)k = n · (n − 1)k-1.

A: To give: A combinatorial proof: For n≥1 and k≥1, k·nk=n·n-1k-1.

Q: Find all natural numbers in such that n²<z". Prove your assertion.

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Q: 7. Prove that 4 3m² – n² for all odd integers m and n.

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A: The function counts the number of coprime integers less than n

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Q: 5.3.2 (a) Suppose that n > 3. Prove that ("+1)ʻ 5.

A: Given:-             n+1n2 &lt; 2 n+1n2 -22&lt;0n+1n+2n+1n-2&lt;0n+1n+2 =…

Q: 4. In(4i)

A: complex number

Q: 5. (*) If a and b are nonzero integers such that a | b and b|a, prove that a = ±b.

A: .

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Q: (2). What is the binary representation of the following: (a) (64.37109375) 10.

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Q: 1) Write all the following as expansions : > 0.2 > V26 V35 (0.99)-4 V(4 – 2y)2 ,y<2 2х-1° 2

A: (a) 0.2answer(a)0.2=2100.2=15(b) (26)13 answer(b) using approximation to solve the cube rootput x=27…

Q: 11. Prove that every positive integer n can be expressed in the form n=4k or n=4k+1 or n=4k+2 or…

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Q: Suppose the product of 22 integers is equal to 1. Prove that their sum is not equal to 0.

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Q: (7) The difference of two odd integers x and y is 4. Prove that x and y are relatively prime.

A: Given difference of two odd integers is 4.

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Q: Is the following statement true or false? n+1 For every odd integer n, To answer this question, let…

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Q: 1. Show for all natural numbers n that (2k – 1) = 1+3+ ...+ (2n – 1) = n². k=1

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Q: 10.2 Find a number a that has the following three properties: (i) a = 73003 (mod 3750). (ii) 1 ≤ a ≤…

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Q: 2.3(i) Find the conjugates, moduli and arguments (to the nearest degree) of the following

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