5 Prove the following statements a For all integers n if 3n

(5) Prove the following statements: (a) For all integers n, if 3n is odd, then n is odd. (b) If 100 coins are distributed over nine bags, then one bag contains at least 12 coins.

Solution

a)

we have to prove

For all integers n, if 3n is odd, then n is odd

here 3 is a odd so n must be odd number because product of two odd integers is odd

we can prove product of two odd integers is odd by followin

Let n and m be two odd integers. By definition of odd we have that n = 2a + 1 and m = 2b + 1.

Consider the product nm

= (2a + 1)(2b +1) = 4ab + 2a + 2b +1=

2( 2ab + a + b) + 1 = 2k + 1, where k = (2ab +a +b ) is an integer.

So by definition of odd we have shown that the product of two odd integers is also odd.

So from this proof we can say that n is odd if 3n is odd

b)

i think you are missing some information in the question.. if it is so then the solution is

Given

100 coins are distributed over nine bags

then one bag contains at least 12 coins. i.e 12 or more than 12 coins

if bag 1 contains 12 coin then rest 88 are distributed among 8 bags

if bag 1 contains 13 coins then rest 87 are distributed among 8 bags

and so on...

if bag 1 contains 92 coins then rest 8 are distributed among 8 bags