13. Suppose n and d are integers and d # 0. Prove each of the following. a. If d |n, then n = [n/d] d. b. If n = [n/d]·d, then d | n. c. Use the floor notation to state a necessary and sufficient condition for an integer n to be divisible by an integer d.

Q: Let n EU = Z and p(n): n² + n is an even integer. a. Write the following statement: (Vn) p(n). b. Is…

A: (a) Here, the given statement is: For ∀n∈U=ℤ, p(n):n2+n is an even integer.

Q: Take any arbitrary integer n and then square it, which one is true? n2 cannot be in the form 7k + 5.…

A: Solution is given as follows :

Q: 3n > n for every positive integer n.

A: Now, since we assumed 3k>k, we can substitute k with 3k in the inequality: 3k+1>3k Since ,…

Q: Let f(sub-n) denote the nth Fibonacci number. Show that f(sub-n) is even if and only if n is…

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Q: * 63. Let a¡, az» ·… , A, be positive real numbers. The arith- metic mean of these numbers is…

A: The arithmetic mean AM=a1+a2+...+annThe geometric mean GM=(a1,a2...,am)1n We need to show A≥G.

Q: Need

A: Given that, a can be written as base 10 x= a0+10a1+ ......+10nar We know that, If a divides b…

Q: Determine whether the following statements are true or false. Justify your answer with a proof or a…

A:

Q: Identify the expression of a for which the following equation holds for every positive integers n…

A:

Q: Write the following expressions in a more compact way (i.e., using index i, j as above). 1. C11 C2x2…

A:

Q: Let m and n be integers. Recall that we say an integer a is even if a=2b for some integer b. We say…

A:

Q: Let n be a positive integer. Write the digits of n in decreasing order, then write the digits in…

A: Sol:- Let the digits of n be d1, d2, ..., dk, where d1 is the highest digit and dk is the lowest…