GRE factorization

The positive integers p and q are both multiples of 3. Which of the following must be a multiple of 6?

2p + 10q + 18 (E). To solve this problem, observe that any multiple of 6 is a multiple of both 2 and 3. Each of the answer choices is a sum of algebraic terms. For the whole expression to necessarily be a multiple of 6, each term must also be a multiple of 6 (i.e. a multiple of both 2 and 3). Let's look at each of the choices: Choice A: Neither p nor q is necessarily a multiple of 2. Choice B: 2p is a multiple of both 2 and 3 but q is not necessarily a multiple of 2. Choice C: 3q is not necessarily a multiple of 2. Choice D: 9 is not a multiple of 2.

Etienne began to eat 20 cookies at exactly the same time Jacques began making more cookies, one at a time, at a constant rate of 16 cookies per hour. If Etienne ate 20 cookies per hour, after how many hours were there no cookies?

5 hours. Etienne and Jacques were working on cross-purposes, so subtract their rates. Usually you add work rates, but this situation is just like a car chase: when one car (or person) gains on another, you subtract rates. 20 cookies per hour − 16 cookies per hour = 4 cookies per hour, so the quantity of cookies decreased by 4 per hour. Since Etienne began with a pile of 20 cookies, it took him 20 ÷ 4 = 5 hours to eat all of the cookies. Note that Etienne ate a lot more than 20 cookies. In 5 hours he ate 100 cookies − the initial 20, plus the 80 that Jacques made in 5 hours.

What is a prime factor?

A factor that is a prime number, stupid.

Modified Work Problem with Multiple Workers: Work=

Individual Rate* Number of Worker * Time

Work=

Rate*Time

Two coal carts, A and B, started simultaneously from opposite ends of a 400-yard track. Cart A traveled at a constant rate of 40 feet per second; Cart B traveled at a constant rate of 56 feet per second. After how many seconds did the two carts collide? (1 yard = 3 feet)

This is a classic combined rates problem. Since the carts are moving directly toward each other, add their rates together. Remember that when two objects are moving in opposite directions - either toward each other or away from each other - add their rates to find how fast the gap is closing (or opening up). To avoid any unit conversion trap (answer choice (E)), do the yards-to-feet conversion up front: 400 yards × 3 ft per yard = 1,200 feet ÷ 96 feet per second = 12 seconds.

Working alone at their respective constant rates, Audrey can complete a certain job in 4 hours, while Ferris can do the same job in 3 hours. Audrey and Ferris worked together on the job and completed it in 2 hours, but while Audrey worked this entire time, Ferris worked for some of the time and took 3 breaks of equal length. How many minutes long was each of Ferris' breaks?

To determine how long Ferris' breaks were, you need to know the difference between the amount of work the two should have completed in two hours, and the amount they actually did complete (that is, one full job). Audrey and Ferris are working together, so first find each worker's individual rate, and then add them together to get the combined rate. Combining the two workers' rates, together they complete 7/12 job per hour, so they should have completed 14/12=7/6 job in two hours. Therefore Ferris' breaks cost them 1/6 worth of productivity. How long was Ferris on break? The amount of time it would have taken him to do 1/6 of the job. At the rate of 1/3 job per hour, Ferris must have spent 1/2 hour on break to miss 1/6 job. Therefore, each of his 3 breaks was 30 minutes ÷ 3 = 10 minutes long. The answer is

Which is larger: The greatest common factor of 2⁵3⁸ and 2⁶3⁹5 or The least common multiple of 2⁴3⁶ and 2⁵3⁷5

To solve this problem, first find the greatest common factor (GCF) of the integers listed in Quantity A and the least common multiple (LCM) of the integers listed in Quantity B. The GCF of the integers listed in Quantity A is the product of each prime factor raised to the smaller of the two powers, or 2⁵3⁸5⁰ = 2⁵3⁸. The LCM of the integers listed in Quantity B is the product of each prime factor raised to the greater of the two powers, or 2⁵3⁷5¹. To compare these values, a calculator is not necessary, since both have 2⁵3⁷ as a common factor; when this is divided away from both, it leaves 3 for Quantity A and 5 for Quantity B. Therefore Quantity B is greater.

distance=

rate*time

For two numbers to have a product that is a multiple of 27

the two numbers need to have at least three 3's among their combined prime factors, since 27 = 3³