1- Following statement is given to you "If m and n are even integers, then mn is divisible by 4." The student used following steps to prove the given statement. If m and n are even integers then there exists integers a and b such that m = 2a andn= 2b Therefore mn can be written mn= (2a)(2b). mn=4ab Since ab is an integer, mn is 4 times an integer so it is divisible by 4. What is the name of the proof method which is used in the solution? a) O Contrapositive proof b) O Direct Proof c) O Mathematical Induction d) O Proof by Contradiction

For each one, state whether the statement is TRUE, FALSE. Then, state the negation:1.∃a(ais a prime number)2.∀a(ais a power of 2)3.∃a∃b(a+b <0)

1. Consider the following statement: "If a number is divisible by 24, then it is divisible by 4 and by 6."e (a) Is this statement true or false?- (b) Write the converse of the above statement. Is this new statement true or false? (c) ) Based on your answers to (a) and (b), is the statement "A number is divisible by 24 if and only if it is divisible by 4 and by 6" true or false? Why?

Prove the 8 divisibility test for ANY number. This should be a mathematical proof, not a word or paragraph proof.    DO NOT write a proof that is for just a four-digit number or a five-digit number.  If you do that then you are not proving it for ANY number.  You must prove it for ANY number by proving that 8 | (10n-1an-1 + 10n-2an-2 + 10n-3an-3 + … + 100a2 + 10a1 + a0).   without modular

Please prove this question the other way around...   10. Is the following proposition true or false? Justify your conclusion. For each integer n, n is even if and only if 4 divides n².   (don't prove what is written above, prove it the other way around please)