M integers

Hi

Statement 1: The odd/even rules are:
- odd X odd = odd
- even x even = even
- even x odd = even

So basically, if the set involved one even number, the product of all numbers will be even. The only way the product of all numbers to be odd is for the set to contain only odd numbers. The number of odd numbers it contains - and also the sign of these numbers - however will not make a difference. so consider the following:

- Set = {-1,-2,-3} -> product = -6 .. so (one even number is enough to make the product even)
- Set = {-1,-3,-5} -> product = -15 ..odd product but odd number of elements
- Set ={-1,-3,-5,-7} -> product = 105.. odd product but even number of elements

Verdict is.. statement 1 is not sufficient, the set size for instance can either be 3 (odd) or 4 (even) and the product will be odd.

Statement 2 tells us that the product of all elements is -ve. The +ve/-ve rules are:
+ve x +ve = +ve
-ve x -ve = +ve
+ve x -ve = -ve

So, the conclusion is that multiplying even number of -ve elements together yields a +ve value. consider the following:

- Set = {-1,-2,-3} -> product = -6...(odd number of elements)
- Set = {-1,-2,-3,-4} -> product = 24...(even number of elements)

So basically, since our case considers only -ve numbers, the only way to get a -ve product is to have an odd number of -ve elements which answers our question.. so statement 2 is sufficient.

statement 1)

oddxodd=odd

in order for product of terms in set to be odd, all terms have to be odd. then we determine if an even set of odd terms is different the odd set of odd terms. answer is no...

oddxodd=odd
odd x odd x odd= odd

insufficient

statement 2)

for product of negative terms in set to be negative there has to be an odd number of terms in set.

negative x negative = positive
negative x negative x negative = negative

therefore M has odd set of terms. sufficient.