How many numbers from 0 to 759 are not divisible by either 3 or 7?

In the June, 2021 Term-End Examination of MTE-13, it was asked to give a direct and an indirect proof of the following statement.

such that a is even and a +b is even, then   b is even."

One student gave an indirect proof as follows:

“Let b be m +1, which is an odd number. We already know a and a +b are even. If we substitute b + m +1 in a + b, it becomes a + m +1, which is an odd number. This contradicts the given statement. Hence b is an even number.”

What is wrong with the above proof ? Also give a correct direct and an indirect proof.

Is the complement of the Peterson graph planar? Justify your answer.b) Is the complement of the Peterson graph planar? Justify your answer.

write down and count all the partitions of the number 7. To verify your answer use the generating function for , Pn taking n= 7 in Theorem 5 (of Unit 5, Block2).

Is the complement of the Peterson graph planar? Justify your answer.

A box contains 6 red and 4 green balls. Four balls are selected from the box at random. What is the probability that two of the selected balls will be red and two will be green? 

  is Hamiltonian, how are m and n related? Justify your answer