Problem Solving

For any integers n greater than 1, n! denotes the product of all the integers from 1 to n, inclusive. How many prime numbers are there between 6! +2 and 6!+6, inclusive?

(A) None
(B) One
(C) Two
(D) Three
(E) Four

Answer is A



I solved this problem by finding each factorial of 6 and add it to 2 and 6 respectively. Is there an easier way to solve this? If the problem asks me to find the number of prime numbers between 1000! + 2 and 1000!+6, then I will be in trouble.

firstly, definition of prime => not more than two factors 1 and the prime itself; secondly, note each factor greater than 2! has more than two factors, hence 6! is not prime; thirdly, you can factor out 2,3,4,5, and 6 --> 2(6*5*4*3*1 +1), 3(6*5*4*2*1+1) ... you see there is always one factor here, hence answer would be none

in case you had between 6! +2 and 6!+8, inclusive, there would be one prime and 8 can be factored as 4*2

cheers

OneTwoThreeFour wrote:For any integers n greater than 1, n! denotes the product of all the integers from 1 to n, inclusive. How many prime numbers are there between 6! +2 and 6!+6, inclusive?

(A) None
(B) One
(C) Two
(D) Three
(E) Four

Answer is A



I solved this problem by finding each factorial of 6 and add it to 2 and 6 respectively. Is there an easier way to solve this? If the problem asks me to find the number of prime numbers between 1000! + 2 and 1000!+6, then I will be in trouble.

OneTwoThreeFour wrote:For any integers n greater than 1, n! denotes the product of all the integers from 1 to n, inclusive. How many prime numbers are there between 6! +2 and 6!+6, inclusive?

(A) None
(B) One
(C) Two
(D) Three
(E) Four

Answer is A

I solved this problem by finding each factorial of 6 and add it to 2 and 6 respectively. Is there an easier way to solve this? If the problem asks me to find the number of prime numbers between 1000! + 2 and 1000!+6, then I will be in trouble.

Just a bit of elaboration on the other explanation.

We know that 6! is a multiple of 2, 3, 4, 5 and 6.

Since 6! is a multiple of 2, (6! + 2) will also be a multiple of 2.
Since 6! is a multiple of 3, (6! + 3) will also be a multiple of 3.
Since 6! is a multiple of 4, (6! + 4) will also be a multiple of 4.
Since 6! is a multiple of 5, (6! + 5) will also be a multiple of 5.
Since 6! is a multiple of 6, (6! + 6) will also be a multiple of 6.

Accordingly, there are no primes between 6!+2 and 6!+6.

For a similar (but much more complicated) question, see:

https://www.beatthegmat.com/gmat-prep-nu ... t8649.html

Thanks! That really helped me clear things up.

@Night reader - Can you please explain this?

in case you had between 6! +2 and 6!+6, inclusive, then there would be no factor here and one prime! if you had between 6! +2 and 6!+8, inclusive, there would be one prime and 8

Night reader wrote:firstly, definition of prime => not more than two factors 1 and the prime itself; secondly, note each factor greater than 2! has more than two factors, hence 6! is not prime; thirdly, you can factor out 2,3,4,5, and 6 --> 2(6*5*4*3*1 +1), 3(6*5*4*2*1+1) ... you see there is always one factor here, hence answer would be none

in case you had between 6! +2 and 6!+6, inclusive, then there would be no factor here and one prime! if you had between 6! +2 and 6!+8, inclusive, there would be one prime and 8 can be factored as 4*2

cheers

OneTwoThreeFour wrote:For any integers n greater than 1, n! denotes the product of all the integers from 1 to n, inclusive. How many prime numbers are there between 6! +2 and 6!+6, inclusive?

(A) None
(B) One
(C) Two
(D) Three
(E) Four

Answer is A



I solved this problem by finding each factorial of 6 and add it to 2 and 6 respectively. Is there an easier way to solve this? If the problem asks me to find the number of prime numbers between 1000! + 2 and 1000!+6, then I will be in trouble.

you should read only "if you had between 6! +2 and 6!+8, inclusive, there would be one prime and 8 can be factored as 4*2 " - the rest was mistypo, I will edit it

for simplicity reason - I go over this post again. If we can factor out any number in the n!+a and n!+b where n,a,b are integers (these question type inquires about primes, so we need to keep only integers here n,a,b), there is no prime. If we can not factor a number likewise above, 7 cannot be factored out, we get prime(s).

crimson2283 wrote:@Night reader - Can you please explain this?

in case you had between 6! +2 and 6!+6, inclusive, then there would be no factor here and one prime! if you had between 6! +2 and 6!+8, inclusive, there would be one prime and 8

Night reader wrote:firstly, definition of prime => not more than two factors 1 and the prime itself; secondly, note each factor greater than 2! has more than two factors, hence 6! is not prime; thirdly, you can factor out 2,3,4,5, and 6 --> 2(6*5*4*3*1 +1), 3(6*5*4*2*1+1) ... you see there is always one factor here, hence answer would be none

in case you had between 6! +2 and 6!+6, inclusive, then there would be no factor here and one prime! if you had between 6! +2 and 6!+8, inclusive, there would be one prime and 8 can be factored as 4*2

cheers

OneTwoThreeFour wrote:For any integers n greater than 1, n! denotes the product of all the integers from 1 to n, inclusive. How many prime numbers are there between 6! +2 and 6!+6, inclusive?

(A) None
(B) One
(C) Two
(D) Three
(E) Four

Answer is A



I solved this problem by finding each factorial of 6 and add it to 2 and 6 respectively. Is there an easier way to solve this? If the problem asks me to find the number of prime numbers between 1000! + 2 and 1000!+6, then I will be in trouble.