(1)
(2M^3 + 2M) / 8 is an integer, doesn't mean much since anything multiplied by an even number is even, so,
2M^3 + 2M =
even*M^3 + even*M =
even + even = even

so (2M^3 + 2M) / 8 => even / even = integer doesn't help us know whether M is odd or even. So (1) is insufficient.

(2) M + 10 is divisible by 10, which must mean

M/10 + 10/10 is an integer.
M/10 + 1, so M/10 must be an integer, so M itself must mean it's a mutiple of 10, which is always even. Sufficient.

Answer is B.

Actually, I take (1) back..

I just tried plugging the values in an excel spreadsheet, and only multiples of 4 for m seems to be divisible by 8, which WOULD allow us to answer that m is definitely not odd, making this answer selection D).

I'm now in the "help wanted" crowd for this problem as well

STMT 1: 2M(M+1) is divisible by 8

For M=4, 2.4.5 is divisible by 8
For M=3, 2.3.4 is also divisible by 8
Hence M can be even or odd and still satisfy stmt 1

Hence insuff

2) M + 10 is divisible by 10
only even numbers are divisible by 10
M+10 must be even. Therefore M must be even.
Hence we get an always NO for the question, Is M odd.
SUfficient

Hence B

My vote for D as well.

1 - sufficient.
2m (m^2 + 1) will be divisible by 8 when m is a multiple of 4.

For odd values of m, (m^2+1) will be divisible by 2 and when
multiplied by 2m, will not be divisible by 8.

So, sufficient to say that m cannot be odd.

2 - sufficient.

I am confused .. B or D

Since 2(M^3) + 2M is divisible by 8 , I can say that this expression is even. In that case M can be odd or even.

But even JayHawk's approach appears right.