Question

Prove that for every integer n, 30 | n iff 5 | n and 6 | n. This problem is similar to examples and exercises in Section 3.4 of your SNHU MAT299 textbook

Answer #1

5. Use strong induction to prove that for every integer n ≥ 6,
we have n = 3a + 4b for some nonnegative integers a and b.

Prove that 7^(n) − 1 is divisible by 6, for every positvie
integer n

Prove that n is prime iff every linear equation ax ≡ b mod n,
with a ≠ 0 mod n, has a unique solution x mod n.

Prove that 2n < n! for every integer n ≥ 4.

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

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

Prove the following theorem: For every integer n, there is an
even integer k such that
n ≤ k+1 < n + 2.
Your proof must be succinct and cannot contain more than 60
words, with equations or inequalities counting as one word. Type
your proof into the answer box. If you need to use the less than or
equal symbol, you can type it as <= or ≤, but the proof can be
completed without it.

Prove the following theorem: For every integer n, there is an even
integer k such that
n ≤ k+1 < n + 2.
Your proof must be succinct and cannot contain
more than 60 words, with equations or inequalities
counting as one word. Type your proof into the answer box. If you
need to use the less than or equal symbol, you can type it as <=
or ≤, but the proof can be completed without it.

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

Prove that for every positive integer n, there exists an
irreducible polynomial of degree n in Q[x].

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