Can someone help me with these questions. 7. Assume that m and n are particular

Question

Can someone help me with these questions.

7. Assume that m and n are particular integers. (So you know that they integers, but you don't know which integers they are.) Answer the following, and justify your answers using the definitions. a. b. c. Is 4m +8n even? Is 24mn +3 odd? If m >n > 1, is m2 - n2 composite? In 8-9, prove the given existential statements: 8. There exist integers m and n such that m > 1 and n > 1 and-+--1. 9. Th d y such that Vr+y-Vr+ v/y In 10-12, disprove the following claims by finding a counterexample: 10. For all integers n, if n 1 then 2n-1 is prime. 11. For all integers m and n, if 4m - n is odd, then m is odd and n is odd. 12. For all real numbers x and y, if x

Explanation / Answer

7) a) Yes 4m+8n is even because 4m is even and 8n is also even (as even number*any integer = even)

and even+even=even

b)Yes 24mn+3 is odd because 24mn is even (as even number*any integer = even)

and even+odd=odd

c) Not necessary that m^2-n^2 will be composite as take m=3 and n=2, then m^2-n^2 = 5 which is prime

8) Let m=1/2 and n=1/2, then m+n=1

9) Let x=1 and y=0, then sqrt(x+y)=sqrt(x)+sqrt(y)

10) Let n=4, then 2^n-1 = 2^4-1 = 16-1 = 15 which is not prime

11) Let m=2 and n=1

then 4m-n= 7 which is odd but m is even

12) Let x=-3 and y=2 so x<y

but x^2 = 9 which is not less than y^2=4

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