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13 Sep 2006, 10:04
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Hello there...
I just got this tough question in one of the GMAT prep test from pearson
What is the least prime factor of the product of the even numbers from 2 to 100 +1?
1) greater than 1 and less than 10.
2) greater than 10 and less than 20.
3) greater than 20 and less than 30.
4) greater than 30 and less than 40.
5) greater than 40.
The answer was greater than 40??
could any one tell how can we get to this answer.
Thanks.....
I just got this tough question in one of the GMAT prep test from pearson
What is the least prime factor of the product of the even numbers from 2 to 100 +1?
1) greater than 1 and less than 10.
2) greater than 10 and less than 20.
3) greater than 20 and less than 30.
4) greater than 30 and less than 40.
5) greater than 40.
The answer was greater than 40??
could any one tell how can we get to this answer.
Thanks.....
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Lest me give you an example
6! = 6*5*4*3*2 = 720
and it is divisible by 5 as largest prime .
then 6!+1 =721 could never be devisible by 5 or 3 or 2 (try it yourself )and thus its least prime factor must be bigger than 720 largest prime factor
There is a rule that if x has y as a prime factor thus sure x+1 doesnt have the same factor y as one of its prime factors and the least prime factor of x+1 must be greater than y.
If we draw the same concept to our problem here
the product of all even numbers till 100 = 2*4......*100
= 2(1*2*4*......47*48*49*50) it is clear that 47 is the largest prime factor of this product
thus for sure the product of even numbers till 100 all + 1
the least prime factor must be greater than 47 thus greater than 40
Hope this helps
6! = 6*5*4*3*2 = 720
and it is divisible by 5 as largest prime .
then 6!+1 =721 could never be devisible by 5 or 3 or 2 (try it yourself )and thus its least prime factor must be bigger than 720 largest prime factor
There is a rule that if x has y as a prime factor thus sure x+1 doesnt have the same factor y as one of its prime factors and the least prime factor of x+1 must be greater than y.
If we draw the same concept to our problem here
the product of all even numbers till 100 = 2*4......*100
= 2(1*2*4*......47*48*49*50) it is clear that 47 is the largest prime factor of this product
thus for sure the product of even numbers till 100 all + 1
the least prime factor must be greater than 47 thus greater than 40
Hope this helps
