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

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Bunuel
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The product of all prime numbers from 2 to 29 plus 1 equals K. Which [#permalink] New post   12 Nov 2019, 03:45
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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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D
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chetan2u
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Re: The product of all prime numbers from 2 to 29 plus 1 equals K. Which [#permalink] New post   12 Nov 2019, 07:17
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Bunuel wrote:
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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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
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Giac
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Re: The product of all prime numbers from 2 to 29 plus 1 equals K. Which [#permalink] New post   14 Nov 2019, 10:50
How can K be odd if it has as prime factor 2?
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chetan2u
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Re: The product of all prime numbers from 2 to 29 plus 1 equals K. Which [#permalink] New post   14 Nov 2019, 18:58
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Giac wrote:
How can K be odd if it has as prime factor 2?


K does not have 2 As a prime factor. K is 2*..+1=even+1=odd
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Re: The product of all prime numbers from 2 to 29 plus 1 equals K. Which [#permalink] New post   16 Nov 2019, 04:03
chetan2u wrote:
Bunuel wrote:
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


Are You Up For the Challenge: 700 Level Questions


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


How can we say it wont be divisble from any prime 2 to 29?
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chetan2u
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Re: The product of all prime numbers from 2 to 29 plus 1 equals K. Which [#permalink] New post   16 Nov 2019, 04:26
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Vinayak013 wrote:
chetan2u wrote:
Bunuel wrote:
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


Are You Up For the Challenge: 700 Level Questions


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


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


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
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Re: The product of all prime numbers from 2 to 29 plus 1 equals K. Which [#permalink] New post   19 Nov 2019, 15:47
Make it simple strategy!

Try with 2, 3, 5.

2*3*5 = 30 + 1 = 31

Hence, just I and II would be correct.
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Re: The product of all prime numbers from 2 to 29 plus 1 equals K. Which [#permalink] New post   22 Feb 2021, 04:23
I think we should also use the concept of co-prime numbers here and that will help us figure out the answer.
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Re: The product of all prime numbers from 2 to 29 plus 1 equals K. Which [#permalink] New post   22 Jul 2022, 23:45
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Re: The product of all prime numbers from 2 to 29 plus 1 equals K. Which [#permalink]
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