(3) Prove that if a cube of a number is divisible by 17, then the number is divisible by 17.

2. Use Fundamental Theorem of Arithmetic to prove that the product of any two odd integers is an odd integer.

Show that the system consisting of all elements of the form a+b√5, where a and b are any rational numbers, satisfies all the axioms for a field if the usual rules for addition and multiplication are used.

Find the negation of the following statement: "Every non-zero integer can be factored as a product of prime numbers." Select one: O a. There exists a non-zero integer that can be factored as a product of prime numbers. O b. All integers cannot be factored as a product of prime numbers. O c. There exists a non-zero integer which cannot be factored as a product of prime numbers. d. All integers can be factored as a product of prime numbers. e. None of the other answers is the negation of this statement. Check

Prove that no number of form 4k+3 (where K is a positive integer) can ever be the sum of two perfect squares. Hint: think about adding two perfect squares.

#17 evaluate the expression

Use contradiction to prove that there is no integer n such that n2 - 2 is divisible by 4.

Prove that if n is an odd number then n 2 + 3 is divisible by 4. Use direct proof.

Show that, given a 7-digit number, where all digits except perhaps the last is non-zero, you can cross out some digits at the beginning and at the end such that the remaining number consists of at least one digit and is divisible by 7. For example, if we take the number 1234589, then we can cross out 1 at the beginning and 89 at the end to get the number 2345 = 7 x 335.

Can anyone prove this question?