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GMAT won't ask you to factor such a huge numbers as \(2^{50}*50!+1\). The stem of the question is enough to answer the question without factoring it. The answer is E (more than 40). _________________
For every positive even integer n, the function h(n) is defnied 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
Any help on how to get the correct answer? Thanks!
Answer: greater than 40
Welcome to the Gmat Club. Below is the solution for your question:
Now, two numbers \(h(100)=2^{50}*50!\) and \(h(100)+1=2^{50}*50!+1\) are consecutive integers. Two consecutive integers are co-prime, which means that they don't share ANY common factor but 1. For example 20 and 21 are consecutive integers, thus only common factor they share is 1.
As \(h(100)=2^{50}*50!\) has all numbers from 1 to 50 as its factors, according to above \(h(100)+1=2^{50}*50!+1\) won't have ANY factor from 1 to 50. Hence \(p\) (\(>1\)), the smallest factor of \(h(100)+1\) will be more than 50.
Answer: More than 50.
Hope it helps.
P.S. Can you please: post one question per topic, tag the questions you post, and also post the whole questions with answer choices.
For every positive even integer n, the function h(n) is defnied 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
Any help on how to get the correct answer? Thanks!
Answer: greater than 40
Welcome to the Gmat Club. Below is the solution for your question:
Now, two numbers \(h(100)=2^{50}*50!\) and \(h(100)+1=2^{50}*50!+1\) are consecutive integers. Two consecutive integers are co-prime, which means that they don't share ANY common factor but 1. For example 20 and 21 are consecutive integers, thus only common factor they share is 1.
As \(h(100)=2^{50}*50!\) has all numbers from 1 to 50 as its factors, according to above \(h(100)+1=2^{50}*50!+1\) won't have ANY factor from 1 to 50. Hence \(p\) (\(>1\)), the smallest factor of \(h(100)+1\) will be more than 50.
Answer: More than 50.
Hope it helps.
P.S. Can you please: post one question per topic, tag the questions you post, and also post the whole questions with answer choices.
Very nice way of solving the problem. Bunuel, you make the problem look easy.
There's no way to prove it without a computer; factoring large integers is one of the most time-consuming mathematical operations (unless they have very obvious factors). The fact that it is so time consuming to factor large numbers is actually the basis of internet encryption systems; you can hack the encryption, but you'd need 10,000 years of computing time to break down the large numbers involved. So Number Theory is the reason you can do banking, shopping or private email on the web. _________________
GMAT Tutor in Toronto
If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com
Cannot understand why according to above h (100)+1 won't have ANY factor from 1 to 50. Let's take the example of 20 & 21; is it true that 21 doesn't have any factor till 20? They won't have common factors like factors for 20 are 2 & 5 while of 21 are 7 & 3, but 21 does have factors less than 20.
Cannot understand why according to above h (100)+1 won't have ANY factor from 1 to 50. Let's take the example of 20 & 21; is it true that 21 doesn't have any factor till 20? They won't have common factors like factors for 20 are 2 & 5 while of 21 are 7 & 3, but 21 does have factors less than 20.
Pl enlighten.
Not sure I understood your question correctly.
Anyway: given that \(p\) is the smallest prime factor of \(h(100)+1=2^{50}*50!+1\). The point here is that consecutive integers do not share ANY common factor but 1. Two numbers \(h(100)=2^{50}*50!\) and \(h(100)+1=2^{50}*50!+1\) are consecutive integers, hence these two integers do not share any common factor but 1. \(h(100)=2^{50}*50!\) naturally has all factors from 2 to 50, which means that \(h(100)+1=2^{50}*50!+1\) does not have any of these factors. So, \(p\), which is the prime factor of \(h(100)+1=2^{50}*50!+1\) must be more than 50. _________________
Anyway: given that \(p\) is the smallest prime factor of \(h(100)+1=2^{50}*50!+1\). The point here is that consecutive integers do not share ANY common factor but 1. Two numbers \(h(100)=2^{50}*50!\) and \(h(100)+1=2^{50}*50!+1\) are consecutive integers, hence these two integers do not share any common factor but 1. \(h(100)=2^{50}*50!\) naturally has all factors from 2 to 50, which means that \(h(100)+1=2^{50}*50!+1\) does not have any of these factors. So, \(p\), which is the prime factor of \(h(100)+1=2^{50}*50!+1\) must be more than 50.
I have got it now. Since h(100) has all the numbers from 2 to 50 as its factor h(100)+1 cannot have any of the numbers between 2 & 50 as its factor. Hence the lowest factor of h(100)+1 is either 1 or a number above 50.
#1 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, the p is? A. btw 2 and 20 B. btw 10 and 20 C. btw 20 and 30 D. btw 30 and 40 E. greater than 40
#2 Pumps A, B, and C operate at their respective constant rates. Pumps A and B, operating simultaneously, can fill a certain tank in 6.5 hours; pumps A and C, operating simultaneously, can fill the tank in 3.2 hours; and pumps B and C, operating simultaneously, can fill the tank in 2 hours. How many hours does it take pumps A,B, and C operating simultaneously, to fill the tank?
#3 Last year certain bond with a face value of $5,000 yielded 8 percent of its face value in interest. If that interest was approximately 6.5 percent of the bond's selling price, approximately what was the bond's selling price? $4,063 $5,325 $5,351 $6,000 $6,154
#1 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, the p is? A. btw 2 and 20 B. btw 10 and 20 C. btw 20 and 30 D. btw 30 and 40 E. greater than 40
Bit Tricky.. The function described will be in the form h(n)= 2^(n/2)*(n/2)! h(100)+1= [(2^50)*50! +1]
By POE we can eliminate ABCD(because any prime number below 50 is a factor of 50!)
#3 Last year certain bond with a face value of $5,000 yielded 8 percent of its face value in interest. If that interest was approximately 6.5 percent of the bond's selling price, approximately what was the bond's selling price? $4,063 $5,325 $5,351 $6,000 $6,154
x- Selling Price 8%5000 = 6.5%x x = (8 * 5000)/6.5 = 6154 hence E.
#1 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, the p is? A. btw 2 and 20 B. btw 10 and 20 C. btw 20 and 30 D. btw 30 and 40 E. greater than 40
Now, two numbers \(h(100)=2^{50}*50!\) and \(h(100)+1=2^{50}*50!+1\) are consecutive integers. Two consecutive integers are co-prime, which means that they don't share ANY common factor but 1. For example 20 and 21 are consecutive integers, thus only common factor they share is 1.
As \(h(100)=2^{50}*50!\) has all numbers from 1 to 50 as its factors, according to above \(h(100)+1=2^{50}*50!+1\) won't have ANY factor from 1 to 50. Hence \(p\) (\(>1\)), the smallest factor of \(h(100)+1\) will be more than 50.
Answer: E.
#2 Pumps A, B, and C operate at their respective constant rates. Pumps A and B, operating simultaneously, can fill a certain tank in 6.5 hours; pumps A and C, operating simultaneously, can fill the tank in 3.2 hours; and pumps B and C, operating simultaneously, can fill the tank in 2 hours. How many hours does it take pumps A,B, and C operating simultaneously, to fill the tank?
I think there is a typos in stem: times should be 6/5 (instead of 6.5) and 3/2 (instead of 3.2).
THEORY If: Time needed for A to complete the job =A hours; Time needed for B to complete the job =B hours; Time needed for C to complete the job =C hours; ... Time needed for N to complete the job =N hours;
Then if time needed for all of them working simultaneously to complete the job is \(T\), then: \(\frac{1}{A}+\frac{1}{B}+\frac{1}{C}+..+\frac{1}{N}=\frac{1}{T}\) (General formula).
For two and three entities (workers, pumps, ...):
General formula for calculating the time needed for two workers A and B working simultaneously to complete one job:
Given that \(a\) and \(b\) are the respective individual times needed for \(A\) and \(B\) workers (pumps, ...) to complete the job, then time needed for \(A\) and \(B\) working simultaneously to complete the job equals to \(T_{(A&B)}=\frac{a*b}{a+b}\) hours, which is reciprocal of the sum of their respective rates (\(\frac{1}{a}+\frac{1}{b}=\frac{1}{t}\)).
General formula for calculating the time needed for three A, B and C workers working simultaneously to complete one job:
\(T_{(A&B&C)}=\frac{a*b*c}{ab+ac+bc}\) hours.
Also for rate problems it's good to know that:
TIME to complete one job=Reciprocal of rate. eg 6 hours needed to complete one job (time) --> 1/6 of the job done in 1 hour (rate).
Time, rate and job in work problems are in the same relationship as time, speed (rate) and distance.
Time*Rate=Distance Time*Rate=Job
Back to our original question. We have: \(\frac{1}{A}+\frac{1}{B}=\frac{1}{t_{a&b}}=\frac{5}{6}\) \(\frac{1}{A}+\frac{1}{C}=\frac{1}{t_{a&c}}=\frac{2}{3}\) \(\frac{1}{B}+\frac{1}{C}=\frac{1}{t_{b&c}}=\frac{1}{2}\)
Sum the first three equations: \(2(\frac{1}{A}+\frac{1}{B}+\frac{1}{C})=\frac{5}{6}+\frac{2}{3}+\frac{1}{2}=2\) --> \(\frac{1}{A}+\frac{1}{B}+\frac{1}{C}=1=\frac{1}{T}\) --> \(T=1\). _________________
first of all thx for all the kind explanations!! it is still confused to understand whole process but i guess i started to get a grip! #1. greater than 40 #2. 1 #3. 6,154
Pumps A, B, and C operate at their respective constant rates. Pumps A and B, operating simultaneously, can fill a certain tank in 6.5 hours; pumps A and C, operating simultaneously, can fill the tank in 3.2 hours; and pumps B and C, operating simultaneously, can fill the tank in 2 hours. How many hours does it take pumps A,B, and C operating simultaneously, to fill the tank?
I think there is a typos in stem: times should be 6/5 (instead of 6.5) and 3/2 (instead of 3.2)
How could you identify that there is a typo dude... ur simply great
Questions from GMAT Prep practice exam (PLEASE HELP!) [#permalink]
03 May 2010, 12:02
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 factior of h (100) + 1, the p is:
* between 2 and 10 * between 10 & 20 * between 20 & 30 * between 30 & 40 * greater than 40
If n is a positive integer and the product of all integers from 1 to n inclusive is a multiple of 990, what is the least possible value of n?
Re: Questions from GMAT Prep practice exam (PLEASE HELP!) [#permalink]
14 May 2010, 03:48
I will tell a more simple and direct solution for this question.
Remember the Wilson's Theorem
p divides (p-1)!+1 if p is prime. The converse is also true.
For instance, 7 is prime, and 7 divides (6!)+1=721
For h(100) = 100!
h(100)+1 = (100!+1)
Which (keeping the Wilson's theorem in mind) can be rewritten as
h(101-1)+1 = [ (101-1)! + 1 ]
101 is the prime which shall divide h(100)+1
and 101 is greater than 40, and choose this answer choice.
Magic lies in the Wilson's theorem, but if you remember the above mentioned method, it will help you in solving any similar or related question without any problem at all.
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