The H.C.F of two numbers is 23 and the other two factors of their L.C.M are 13 and 14. The larger of the two numbers is:
Options:
(A) 276
(B) 299
(C) 322
(D) 345
(E) 355
Answer and stream of consciousness: (C)
Stream of consciousness: Clearly, the numbers are (23 * 13) and (23 * 14). Larger number = (23 * 14) = 322.

There are 39 distinct numbers in set M, there are 28 distinct numbers in set N, and there are 12 distinct numbers that are in both sets M and N. Set H is the set containing the elements that are in at least one of sets M and N. How many elements are in set H?
Options:
(A) 39
(B) 40
(C) 55
(D) 58
(E) 63
Answer and stream of consciousness: (C)
Stream of consciousness: {Total} = {M} + {N} - {Both}
{Total} = 39 + 28 - 12 = 55.

The average of 13 numbers is 60. Average of the first 7 of them is 57 and that of the last 7 is 61. Find the 8th number?
Options:
(A) 46
(B) 77
(C) 88
(D) 544
(E) 33
Answer and stream of consciousness: (A)
Stream of consciousness: Sum of all the 13 numbers = 13 * 60 = 780
Sum of the first 7 of them = 7 * 57 = 399
Sum of the last 7 of them = 7 * 61 = 427
So, the 8th number = 427 + 399 - 780 = 46.

If x<0, y>0, and |x| < |y|, which of the following must be true?
Options:
(A) x > y
(B) y^2 > x^2
(C) x^3 > y^2
(D) x < y
(E) x < y
Answer and stream of consciousness:
(D)
Stream of consciousness: Correct Answer: (E)
Let's go through each answer choice: (A) can never be true, since no negative is greater than a positive. (B) doesn't have to be true consider what would happen if x = -2 and y = 1. (C) can never be true, as x^3 must be negative, and y^2 must be positive. (D) can never be true, since if x < 0, -x is the same thing as |x|, and |x| > y. (E) can be manipulated by multiplying both sides by -1, which gives us x > y. Remember that x < 0, so x = |x|, and y is positive, so |y| = y. Thus x < y is the same statement as |x| > |y|, and (D) must be true.