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Product of consequtive integers problem

Expert replies

by Strongt » Sat Apr 23, 2011 11:06 am
I know Im bringing this topic from the dead, but i have a question.

Why can't 10 be the least value?

or does since the prime number?
2 and 5 are contained 10. and 10 990 can be divided by 10

I know I'm missing something here but I really want to master this question type
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by Stuart@KaplanGMAT » Sat Apr 23, 2011 11:42 am
Strongt wrote:I know Im bringing this topic from the dead, but i have a question.

Why can't 10 be the least value?

or does since the prime number?
2 and 5 are contained 10. and 10 990 can be divided by 10

I know I'm missing something here but I really want to master this question type
Hi!

Our number has to be a multiple of 990. The only way it could possibly be multiple of 990 is if it contains all of the prime factors of 990.

Another way to phrase the question is:

if n!/990 is an integer, what's the smallest possible value of n?

So, 990 must divide into n!, which means that n! must contain all the prime factors of 990.

Since 990 = 99*10 = 9*11*10, and since 11 is prime (i.e. can't be broken down any further), 11! is the smallest factorial that's divisible by 990; accordingly, 11 is the smallest possible value of n.
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by jsche229 » Wed Jun 04, 2014 8:52 am
I know this is going back a bit but why wouldn't 10 work (2*5)??? Thanks!
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by confused13 » Tue Jul 01, 2014 11:39 am
Is this question, worth answering?
I did calculate the prime factors but honestly I didn't know what to do with them.
So is this a must-get question?
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by Brent@GMATPrepNow » Tue Jul 01, 2014 12:14 pm
confused13 wrote:Is this question, worth answering?
I did calculate the prime factors but honestly I didn't know what to do with them.
So is this a must-get question?
Yes, I'd say that the concepts involved here are pretty important to know for test day.

Cheers,
Brent
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by Brent@GMATPrepNow » Tue Jul 01, 2014 12:56 pm
confused13 wrote:Is this question, worth answering?
I did calculate the prime factors but honestly I didn't know what to do with them.
So is this a must-get question?
By the way, here's my solution to this question:
If n is a positive integer and the product of all the integers from 1 to n, inclusive, is a multiple of 990. What is least possible value of n?
a) 10
b) 11
c) 12
d) 13
e) 14
A lot of integer property questions can be solved using prime factorization.
For questions involving divisibility, divisors, factors and multiples, we can say:
If N is divisible by k, then k is "hiding" within the prime factorization of N
Similarly, we can say:
If N is is a multiple of k, then k is "hiding" within the prime factorization of N

Examples:
24 is divisible by 3 <--> 24 = 2x2x2x3
70 is divisible by 5 <--> 70 = 2x5x7
330 is divisible by 6 <--> 330 = 2x3x5x11
56 is divisible by 8 <--> 56 = 2x2x2x7

So, if if some number is a multiple of 990, then 990 is hiding in the prime factorization of that number.

Since 990 = (2)(3)(3)(5)(11), we know that one 2, two 3s, one 5 and one 11 must be hiding in the prime factorization of our number.

For 11 to appear in the product of all the integers from 1 to n, n must equal 11 or more.
So, the answer is B

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by GMATinsight » Tue Jul 01, 2014 8:10 pm
990 = 2x3x3x5x11

All the factors of 990 will be part of 11! (11 factorial), which is product of all integers from 1 till 11

Therefore answer Option: B
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