Miscellaneous

07/11/05 Question What is the value of (a! + b!)(c! + d!)? (1) b!d! = 4(a!d!) (2) 60(b!c!) = (b!d!) (A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not. (B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not. (C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient. (D) EACH statement ALONE is sufficient to answer the question. (E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question. Answer There is no useful rephrase of the question, so the best approach here is to analyze the statement to see what they tell us about the values of a, b, c, and d. Statement 1 tells us that b!d! = 4(a!d!). If we divide both sides by d!, we are left with b! = 4a!. Remember that to find the factorial value of an integer, you multiply that integer by every positive integer smaller than it. Since b! is 4 times greater than a!, it must be true that b! = 4 x a x (a - 1) x (a - 2)... Since b! is a factorial product and cannot have more than one 4 as a factor, it must be true that b! = 4 x 3 x 2 x 1. Therefore, a = 3 and b = 4. But this tells us nothing about c or d. Insufficient. Statement 2 tells us that 60(b!c!) = (b!d!). If we divide both sides by b!, we are left with 60c! = d!. Since d! is 60 times greater than c!, d! could equal 60! (i.e., 60 x 59 x 58...), and therefore d = 60 and c = 59. Or d! could equal (c!)(3)(4)(5), in which case c! must be 2! and c = 2 and d = 5. Insufficient. If we pool the information from both statements, however, we see that 60(b!c!) = 4(a!d!), which yields 15(b!c!) = (a!d!). If we try this equation with a = 3, b = 4, c = 59, and d = 60, we get 15(4!59!) = (3!60!) or 60(3!59!) = (3!60!), which is the same as 3!60! = 3!60!. So these four values are possible. If we try the equation with a = 3, b = 4, c = 2, and d = 5, we get 15(4!2!) = (3!5!) or (3)(5)(4!2!) = (3!5!), which is the same as 5!3! = 3!5!. So these four values are possible as well. Since the value of c can be either 2 or 59 and the value of d can be either 5 or 60, we cannot answer the question definitively. The correct answer is E. 03/07/05 Question

If

, what is the value of

in terms of

?

(A) (B) (C) (D) (E) 1 Answer The simplest approach to this problem is to pick numbers. Let's say that x = 1. We can plug in 1 for x in

And we can plug in 1 for x in

:

:

Therefore, give

will be equal to

. If we evaluate each choice by plugging in 1 for x, the only one to

as an answer is A:

Alternatively, we can solve algebraically.

First, let's calculate the value of

:

. We can simplify this in terms of

Now let's calculate the value of

:

. We can simplify this in terms of

We can now see that

is equal to the following:

:

:

The correct answer is A. 02/03/03 Question

If

, and a and b are both non-zero integers, which of the following could be the value of b?

I. 2 II. 3 III. 4 (A) I only (B) II only (C) I and II only (D) I and III only (E) I, II, and III Answer First, simplify the numerator by letting x = ab. Then the numerator can be simplified as follows:

Substituting ab back in for x, the original equation now looks like this:

In order for the fraction to have a value of 0, the numerator must have a value of 0. Thus, ab can be equal to 0, -6, or 3. However, since we are told that a and b are both nonzero integers, ab cannot be 0 and it must be equal to -6 or 3. Therefore, a and b must be integer factors of -6 or 3. Thus it would appear that: b can be equal to 2, if a = -3 b can be equal to 3, if a = -2 b can be equal to 3, if a = 1 However, a cannot be equal to -2 or 1, since this would make the denominator equal to 0 and leave the fraction undefined. This leaves one option: a = -3 and b = 2. The correct answer is A (I only): the variable b can be equal to 2, not 3 or 4.