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distance-time and prime factor

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distance-time and prime factor

by Senthilnathan Vedi » Sat Oct 13, 2012 2:22 am
Hi All,
I got these two questions in one my GMATPrep practice tests. Can anyone please give the explanations for the official answers.

Q18:
------
A boat traveled upstream a distance of 90 miles at an average speed of
(v-3) miles per hour and then traveled the same distance downstream at an
average speed of (v+3) miles per hour. If the trip upstream took half an
hour longer than the trip downstream, how many hours did it take the boat
to travel downstream?

a) 2.5
b) 2.4
c) 2.3
d) 2.2
e) 2.1

[spoiler]OA: A[/spoiler]

Q27:
------
For every positive even integer n, the function h(n) is defined to be the
product of all the even integers from 2 to n,inclusive. If 'p' is the
smallest prime factor of h(100) + 1, then 'p' is

(a) between 2 and 10
(b) between 10 and 20
(c) between 20 and 30
(d) between 30 and 40
(e) greater than 40

[spoiler]OA: E[/spoiler]
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by Anurag@Gurome » Sat Oct 13, 2012 2:33 am
Senthilnathan Vedi wrote:Hi All,
I got these two questions in one my GMATPrep practice tests. Can anyone please give the explanations for the official answers.

Q18:
------
A boat traveled upstream a distance of 90 miles at an average speed of
(v-3) miles per hour and then traveled the same distance downstream at an
average speed of (v+3) miles per hour. If the trip upstream took half an
hour longer than the trip downstream, how many hours did it take the boat
to travel downstream?

a) 2.5
b) 2.4
c) 2.3
d) 2.2
e) 2.1

[spoiler]OA: A[/spoiler]
Time taken to travel upstream = 90/(v - 3) hr
Time taken to travel downstream = 90/(v + 3) hr

Traveling upstream took 1/2hr more than downstream.
So, 90/(v - 3) = 90/(v + 3) + 1/2
180(v + 3) = 180(v - 3) + (v² - 9)
1080 = v² - 9 or v = √1089 = 33

Speed for downstream = 33 + 3 = 36 mph
Therefore, time taken to travel downstream = 90/36 = 2.5 hrs

The correct answer is A.
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by Anurag@Gurome » Sat Oct 13, 2012 2:34 am
Senthilnathan Vedi wrote:Hi All,
I got these two questions in one my GMATPrep practice tests. Can anyone please give the explanations for the official answers.

Q27:
------
For every positive even integer n, the function h(n) is defined to be the
product of all the even integers from 2 to n,inclusive. If 'p' is the
smallest prime factor of h(100) + 1, then 'p' is

(a) between 2 and 10
(b) between 10 and 20
(c) between 20 and 30
(d) between 30 and 40
(e) greater than 40

[spoiler]OA: E[/spoiler]
h(100) = 2 * 4 * 6 * ... * 100
= (2 * 1) * (2 * 2) * (2 * 3) * ... * (2 * 50)
= 2^(50) * (1 * 2 * 3 ... * 50)
Then h(100) + 1 = 2^(50) * (1 * 2 * 3 ... * 50) + 1
Now, h(100) + 1 cannot have any prime factors 50 or below, because dividing this value by any of these prime numbers will give a remainder of 1.

The correct answer is E.
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by Senthilnathan Vedi » Sat Oct 13, 2012 3:58 am
Thank you Anurag.
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by GMATGuruNY » Sat Oct 13, 2012 4:08 am
Senthilnathan Vedi wrote: A boat traveled upstream a distance of 90 miles at an average speed of
(v-3) miles per hour and then traveled the same distance downstream at an
average speed of (v+3) miles per hour. If the trip upstream took half an
hour longer than the trip downstream, how many hours did it take the boat
to travel downstream?

a) 2.5
b) 2.4
c) 2.3
d) 2.2
e) 2.1
We can PLUG IN THE ANSWERS, which represent the number of hours that the boat took to travel downstream.
The most likely answer choice is A -- the only option that divides evenly into 90.

Answer choice A: 2.5 hours to travel downstream.
Rate downstream = d/t = 90/(2.5) = 36 miles per hour.
Thus, v+3 = 36, implying that v=33.
Rate upstream = v-3 = 33-3 = 30 miles per hour.
Time upstream = d/r = 90/30 = 3 hours.
Time upstream - time downstream = 3-2.5 = .5.
Success!

The correct answer is A.

Please note the following:

-- Because we assessed the viability of the answer choices BEFORE we started to plug in, we had to try only ONE answer choice.

-- Plugging in the answers enabled us to determine the correct answer -- quickly and efficiently -- without having to set up and solve a quadratic equation.
Last edited by GMATGuruNY on Sun May 11, 2014 4:37 pm, edited 1 time in total.
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by GMATGuruNY » Sat Oct 13, 2012 4:10 am
For every positive even integer n, the function h(n) is defined to be the product of all even integers from 2 to n, inclusive. If p is the smallest prime factor of h(100) + 1, then p is
Options:
A: Between 2 & 10
B: Between 10 & 20
C: Between 20 & 30
D: Between 30 & 40
E: Greater than 40
Since the difference between them is 1, h(100) and h(100)+1 are consecutive integers. Consecutive integers are co-primes: they share no factors other than 1.

Let's examine why:

If x is a multiple of 2, the next largest multiple of 2 is x+2.
If x is a multiple of 3, the next largest multiple of 3 is x+3.

Using this logic, if we go from x to x+1, we get only to the next largest multiple of 1. So 1 is the only factor common both to x and to x+1. Integers that share no factors other than 1 are called coprimes.

Thus, in the problem above, h(100) and h(100)+1 are coprimes. They share no factors other than 1.

h(100) = 2 * 4 * 6 *....* 94 * 96 * 98 * 100

Factoring out 2, we get:

h(100) = 2^50 (1 * 2 * 3 *... * 47 * 48 * 49 * 50)

Looking at the set of parentheses on the right, we can see that every prime number between 1 and 50 is a factor of h(100). Since h(100) and h(100)+1 are coprimes, none of the prime numbers between 1 and 50 can be a factor of h(100)+1.

Thus, the smallest prime factor of h(100) + 1 must be greater than 50.

The correct answer is E.
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