**#25**

Let x and y be positive integers such that 3x + 7y is divisible by 11. Which of the following is true?

a. 4x + 6y

b. x + y + 5

c. 9x + 4y

d. 4x - 9y

e. x + y -1

The answer is d. 4x - 9y

My logic:

Since x and y are positive integers and 3 and 7 are relative primes, then 11 | 3x + 7y would imply that 11 | x and 11 | y. Thus, any linear combination of x and y should be divisible by 11.

I know my logic is wrong, can anyone tell me what's wrong?

**#30**

The improper integral, from b to a, f(x)*f'(x) dx is

The answer is a. necessarily zero.

My logic:

If I let f(x) = x^2, then f'(x) = 2x, meaning f(x)f'(x) = 2x^3. The integral, from b to a, 2x^3 dx is 1/2 * (b^4 - a^4), which is not zero.

I know my logic is wrong, can anyone tell me what's wrong?

**#50**

In a game two players take turns tossing a fair coin, the winner is the first one to toss a head. The probability that the player who makes the first toss wins the game is?

The answer is D: 2/3.

My logic:

Let the two player be called A and B. Suppose A go first, and toss a head, the game is done and the probability of this event is 1/2. Suppose A go first and does not toss a head, then B's turn, and toss a head, the game is done, and the probability of this event is 1/2 * 1/2. Since the name A and B are generic, the sum of the probability of these two disjoint events should be the answer, which is 3/4.

I know my logic is wrong, can anyone tell me what's wrong?