Prove that for any integer n the expression n7/7 + n5/5 + 23n/35 is whole integer....

Prove that for any integer n the expression
n7/7 + n5/5 + 23n/35
is whole integer.
(Hint: Note that the problem can be state in a following equivalent form: 35 | (5n7 +7n5 +23n);
even further, by the previews theorem, it would be enough to show that (5n7 + 7n5 + 23n) is
divisible by 5 and 7.)

Expert Solution

Colby Messinger answered 2 years ago

Prove that τ(n) < 2 n for any positive integer n. This is a question in...

Prove that τ(n) < 2 n for any positive integer n. This is a question in Number theory

Use induction to prove that for any positive integer n, 8^n - 3^n is a multiple...

Use induction to prove that for any positive integer n, 8^n - 3^n is a multiple of 5.

3.11. (a) Let n be any integer such that n is congruent to 0 (mod 7)....

3.11. (a) Let n be any integer such that n is congruent to 0 (mod 7). For any positive integer k, what is the remainder when n^k is divided by 7? (b) Let n be any integer such that n is congruent to 1 (mod 7). For any positive integer k, what is the remainder when n^k is divided by 7? (c) Let n be any integer such that n is congruent to 2 (mod 7). For any nonnegative integer...

prove or disprove .if n is a non negative integer, then 5 divides 2 ⋅ 4^n...

prove or disprove .if n is a non negative integer, then 5 divides 2 ⋅ 4^n + 3⋅9^n.

Prove that if n is an integer and n^2 is even the n is even.

Let n be a positive integer. Prove that if n is composite, then n has a...

Let n be a positive integer. Prove that if n is composite, then n has a prime factor less than or equal to sqrt(n) . (Hint: first show that n has a factor less than or equal to sqrt(n) )