Question

Suppose a is an integer. Prove that if a 2 is a multiple of 3, then a is a multiple of 3

Answer #1

Prove that there is no integer which is a perfect square and is
a multiple of 2, but is not a multiple of 4.

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

Prove that if a is an integer, then 3 | (a^3 - a).

Prove that for every positive integer n, there exists a multiple
of n that has for its digits only 0s and 1s.

Suppose n ≥ 3 is an integer. Prove that in Sn every
even permutation is a product of cycles of length 3.
Hint: (a, b)(b, c) = (a, b, c) and (a, b)(c, d) = (a, b, c)(b,
c, d).

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 that for every integer m, the number (m^3 +3m^2 +2m)/6 is
also an integer
can I get a step by step induction proof please.

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.

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.

Suppose for each positive integer n, an is an integer such that
a1 = 1 and ak = 2ak−1 + 1 for each integer k ≥ 2. Guess a simple
expression involving n that evaluates an for each positive integer
n. Prove that your guess works for each n ≥ 1.
Suppose for each positive integer n, an is an integer such that
a1 = 7 and ak = 2ak−1 + 1 for each integer k ≥ 2. Guess a...

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