Question

let n be an odd integer ,prove that 5460 | n^25-n

Answer #1

Let n be an odd integer.
Prove that 5460 | n25 −n

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.

Let n be a positive odd integer, prove gcd(3n, 3n+16) = 1.

Let n be an integer greater than 2. Prove that every subgroup of
Dn with odd order is cyclic.

(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?

6. Consider the
statment. Let n be an integer. n is odd if and
only if 5n + 7 is even.
(a) Prove the forward implication of this statement.
(b) Prove the backwards implication of this statement.
7. Prove the following statement. Let a,b, and
c be integers. If a divides bc and
gcd(a,b) = 1, then a divides c.

Prove the following: Let n∈Z. Then n2 is odd if and
only if n is odd.

Let n be any integer, prove the following statement:
n3+ 1 is even if and only if n is odd.

Let m = 2k + 1 be an odd integer. Prove that k + 1 is the
multiplicative inverse of 2, mod m.

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.]

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