Question

Show that n = ∑ d ∣ n ϕ ( d ) for all positive integers n.

Answer #1

Show that if a and d are positive integers, then (-a) div d= -a
div d if and only if d divides a.

1. A) Show that the set of all m by n matrices of integers is
countable where m,n ≥ 1 are some ﬁxed positive integers.

Show that for all positive integers n
∑(from i=0 to n) 2^i=2^(n+1)−1
please use induction only

Characterize the set of all positive integers n for which φ(n)
is divisible by 2 but not by 4

Show that the set of all functions from the positive integers to
the set {1, 2, 3} is uncountable.

Prove that for all positive integers n,
(1^3) + (2^3) + ... + (n^3) = (1+2+...+n)^2

1. Let n be an odd positive integer. Consider a list of n
consecutive integers.
Show that the average is the middle number (that is the number
in the
middle of the list when they are arranged in an increasing
order). What
is the average when n is an even positive integer instead?
2.
Let x1,x2,...,xn be a list of numbers, and let ¯ x be the
average of the list.Which of the following
statements must be true? There might...

Prove that for fixed positive integers k and n, the number of
partitions of n is equal to the number of partitions of 2n + k into
n + k parts.
show by using bijection

Problem 1. Prove that for all positive integers n, we have 1 + 3
+ . . . + (2n − 1) = n ^2 .

Find all integers n (positive, negative, or zero) so
that (n^2)+1 is divisible by n+1.
ANS: n = -3, -2, 0, 1

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