Question

Prove that if n is an integer and 3 is a factor of n 2 , then 3 is a factor of n.

Answer #1

3.a) Let n be an integer. Prove that if n is odd, then
(n^2) is also odd.
3.b) Let x and y be integers. Prove that if x is even and y is
divisible by 3, then the product xy is divisible by 6.
3.c) Let a and b be real numbers. Prove that if 0 < b < a,
then (a^2) − ab > 0.

Prove every integer n ≥ 2 has a prime factor. (You cannot just
cite the Funda- mental Theorem of Arithmetic; this was the first
step in proving the Fundamental Theorem of Arithmetic

Prove or disprove that 3|(n^3 − n) for every positive integer
n.

a) Prove: If n is the square of some integer, then n /≡ 3 (mod
4). (/≡ means not congruent to)
b) Prove: No integer in the sequence 11, 111, 1111, 11111,
111111, . . . is the square of an integer.

(a) Let N be an even integer, prove that GCD (N + 2, N) = 2.
(b) What’s the GCD (N + 2, N) if N is an odd integer?

Let n be an integer, with n ≥ 2. Prove by contradiction that if
n is not a prime number, then n is divisible by an integer x with 1
< x ≤√n.
[Note: An integer m is divisible by another integer n if there
exists a third integer k such that m = nk. This is just a formal
way of saying that m is divisible by n if m n is an integer.]

Show 2 different solutions to the task.
Prove that for every integer n (...-3, -2, -1, 0, 1, 2, 3,
4...), the expression n2 + n will always be even.

Prove that every integer of the form 5n + 3 for n ∈ Z, n ≥ 1,
cannot be a perfect square

Prove let n be an integer. Then the following are
equivalent.
1. n is an even integer.
2.n=2a+2 for some integer a
3.n=2b-2 for some integer b
4.n=2c+144 for some integer c
5. n=2d+10 for some integer d

Prove that for each positive integer n, (n+1)(n+2)...(2n) is
divisible by 2^n

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