The product of all prime numbers from 2 to 29 plus 1 equals K. Which

Question

The product of all prime numbers from 2 to 29 plus 1 equals K. Which of the following is true?

I. K is divisible by prime number.
II. K has a prime factor greater than 29.
III. K is divisible by 30

A. I only
B. II only
C. III only
D. I and II only
E. I, II and III

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chetan2u · Accepted Answer

K=2*3*5*7*11*13*17*19*23*29+1.....K is ODD
Surely K will NOT be divisible by any of the prime number from 2 to 29

I. K is divisible by prime number.
 All numbers>1 are surely divisible by some prime number..True

II. K has a prime factor greater than 29.
 K is NOT divisible by any of the prime factors from 2 to 29, so it has to have a prime factor>29..True

III. K is divisible by 30
 K is ODD..False

I and II true

D

Giac · Answer

How can K be odd if it has as prime factor 2?

chetan2u · Answer

K does not have 2 As a prime factor. K is 2*..+1=even+1=odd

Vinayak013 · Answer

How can we say it wont be divisble from any prime 2 to 29?

chetan2u · Answer

Now say x=2*3*4..*29 is product of all numbers from 2 to 29, so x is divisible by all numbers from 3 to 29.
But the moment you add 1 to x, that is x+1 will NOT be divisible by any numbers that x is divisible by because x and x+1 are consecutive numbers. 
So x+1 is NOT divisible by any numbers from 2 to 29. And x+1=2*3*4..*29+1=k

jfranciscocuencag · Answer

Make it simple strategy!

Try with 2, 3, 5.

2*3*5 = 30 + 1 = 31

Hence, just I and II would be correct.

aaditya005 · Answer

I think we should also use the concept of co-prime numbers here and that will help us figure out the answer.

jacob46 · Answer

Hi Chetan,

How do we also assume that K here in itself will not be a prime factor? If K is prime it wouldn't have any prime factors right?

Thank you

GMAT.IQ · Answer

K may or may not be a prime number.  (It actually isn't a prime number.)  However, it doesn't matter.  Even if K is a prime number, it is divisible by 1 and itself, which is a prime number.  This is also why "I." was true.

GMAT.IQ · Answer

There is no way to  definitely  know that K won't be divisble from any prime number from 2 to 29 unless you are a math major as this involves number theory, which isn't taught in high school.  The following is the proof if you're interested.  We know that K must be a prime number because K is not divisible by 2.  We know this because K - 1 is divisible by 2.  And only numbers that you add 2 to will be divisible.  So K - 1, K + 1, K + 3, K + 5, etc., are all divisible by 2.  K, K + 2, K + 4, K +6, etc., are not divisible.

Similarly, K is not divisible by 3 because K -1 is divisible by 3.  So only K - 1, K + 2, K + 5, K + 8, etc., are divisible by 3.  And K, K + 1, K + 3, K + 4, etc., are not divisible by 3.

Similarly, K is not divisible by 5 because K - 1 is divisible by 5, so K + 1, K + 2, K + 3, and K + 4 are not divisible by 5.  K + 5, on the other hand,  is  divisible by 5.

You follow the same logic with all the prime numbers up to 29, so none of them are factors of K.  So the prime factor that is a factor of K must be bigger than 29.

You obviously can't think all of this on the test, so what are you supposed to do?  I think you have to go by a general guess.  (This is why I don't like this question.)  Chechen states that "x+1 will NOT be divisible by any numbers that x is divisible by because x and x+1 are consecutive numbers".  This is true, but I don't know if you can just blindly accept this.  What you would need to do is think of each prime number and consider when it's possible that K is divisible by that number.  And test out the first few such as 2, 3, 5, and 7.

crimson_king · Answer

A shortcut way to get the answer within a minute

Since we are calculating the product of all the prime numbers from 2 to 29 & adding 1 to the resulting value, we can affirm that the resulting number will be an odd number. This automatically negates the third option as the integral multiples of 30 cannot ever be odd numbers. Hence options (C) & (E) are out

Now for the remaining 2 options, we are presented with a bit of a conundrum. We know that option (B) cannot be by itself true without option (A) being true, hence we can effectively eliminate option (B), which leaves us with options (A) & (D)

Now if we take the case of numbers 2&3 & add 1 to the resulting product we find that the number is divisible by 7 which is a prime number which lies outside the range of the numbers used (in this case 2&3). Extrapolating that for the product of prime numbers from 2 to 29, we find that it can only be divisible by a prime number above 29 & by default above 30, hence option (A) can be eliminated & we get option (D) as our answer

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GMAT.IQ · Answer

You stated that since 7 is a prime number, you can extrapolate that the prime number has to be above 29.  Although this is true, it's a leap in logic.  Your reason isn't exactly true.  It's the fact that you need to think about each prime number individually.  A multiple of a prime number + 1 can't be a multiple of that prime number.

I do agree though that if you multiply a first few of the prime numbers, you get the sense that the resulting number plus 1 can't be divisible by any of the prime numbers.  With that you, conclude that II is not possible.  However, this is more a vague guess, thus I have an issue with this problem.

nihilre · Answer

how to know if its divisible by any prime number

Bunuel · Answer

The source of the question is unknown.

The product of all prime numbers from 2 to 29 plus 1 equals K. Which

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The product of all prime numbers from 2 to 29 plus 1 equals K. Which

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