pallaviisinhaStatement 1: 2M^3 + 2M mod 8 = 0 -----> 2M(M^2+1) mod 8 =0 -----> M(M^2+1) mod 4 = 0

We know that square of even is even and square of odd is odd. So, if M is even, (M^2+1) will be odd (as even*even+1 is odd)

and if M is odd (M^2+1) will be even (as odd*odd +1 will be even.

For M(M^2+1) mod 4 = 0 to be true we want one of M or (M^2+1) to be divisible by 4.

For even M, such as M=4, we can have M(M^2+1) mod 4 = 0

For any odd M... (M^2+1) mod 2 will be = 0 but (M^2+1) mod 4 will never be = 0

For ex. M=3 ---> M^2+1=10, M=5 ---> M^2+1=26, M=7 ---> M^2+1=50, M=9 ----> M^2+1=82

[all (M^2+1) mod 4 is not equal to 0]Hence, we can conclude that M has to be even in order to have M(M^2+1) mod 4 = 0

Therefore, Statement 1 is sufficient.

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P.S. Don't forget to give kudos if you got the explanation.

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P.S. Don't forget to give Kudos