For an integer n greater than 1, n* denotes the product

Question

For an integer n greater than 1, n* denotes the product of all the integers from 1 to n inclusive. How many prime numbers are there between 7*+2 and 7*+7, inclusive ?

A. 0
B. 1
C. 2
D. 3
E. 4

BrentGMATPrepNow · Accepted Answer

7* = 7! = (7)(6)(5)(4)(3)(2)(1)

So, 7* + 2 = (7)(6)(5)(4)(3)( 2 )(1) +  2 
=  2  
= some multiple of  2 
So, 7* + 2 is NOT a prime number

7* + 3 = (7)(6)(5)(4)( 3 )(2)(1) +  3 
=  3  
= some multiple of  3 
So, 7* + 3 is NOT a prime number

7* + 4 = (7)(6)(5)( 4 )(3)(2)(1) +  4 
=  4  
= some multiple of  4 
So, 7* + 4 is NOT a prime number
.
.
.
.
7* + 7 = ( 7 )(6)(5)(4)(3)(2)(1) +  7 
=  7  
= some multiple of  7 
So, 7* + 7 is NOT a prime number

ASIDE: You can assume that I was able to perform the same steps with 7* + 5 and 7* + 6

Answer: A

Cheers,
Brent

Bunuel · Answer

Given that n* denotes the product of all the integers from 1 to n, inclusive so, 7* + 2 = 7! + 2 and 7* + 7 = 7! + 7. Now, notice that we can factor out 2 our of 7! + 2 so it cannot be a prime number, we can factor out 3 our of 7! + 3 so it cannot be a prime number, we can factor out 4 our of 7! + 4 so it cannot be a prime number, ... The same way for all numbers between 7! + 2 and 7! +7, inclusive. Which means that there are no primes in this range. Answer: A. Similar questions to practice: for-any-integer-n-greater-than-1-n-denotes-the-product-of-168575.html for-any-integer-m-greater-than-1-m-denotes-the-product-of-148722.html for-any-integer-p-p-is-equal-to-the-product-of-all-the-int-112494.html for-any-integer-n-greater-than-1-n-denotes-the-product-of-131199.html for-any-integer-k-greater-than-1-the-symbol-k-denotes-the-105890.html does-the-integer-k-have-a-factor-p-such-that-1-p-k-126735.html if-x-is-an-integer-does-x-have-a-factor-n-such-that-100670.html

Lucky2783 · Answer

None. as 7! is multiple of all numbers in the set {1,2,3,4,5,6,7} so when you add any of these number to the 7! you will always a multiple of one the numbers in the set. similar question : does-the-integer-k-have-a-factor-p-such-that-1-p-k-126735.html

gmatkiller88 · Answer

Hi Bunuel,

Based on the concept you mentioned that we can take a factor out from each number which results in no prime numbers between 7!+2 and 7!+7, can we say that if we add to 7! a number which is not a factor of 7* then the resultant number will be prime no.?

e.g. can we say 7!+11 or 7!+13 are prime numbers? Thanks in advance.

shreyast · Answer

Hi  gmatkiller88 ,

Not necessarily so. Here is a counter example, 3! + 19 = 25, not prime.

gmatkiller88 · Answer

Ok. So I guess in such cases we need to check each number individually to decide if the number is prime or not. Thanks.

mahakmalik · Answer

I did it this way
7!+2=2(7*6*5*4*3*1+1)=this will not be prime
7!+3=3(7*6*5*4*2*1+1)=this will not be prime
.......
7!+7=7(7*6*5*4*2*1+1)=same,not prime as it is multiple of 7
answer=0/A

ScottTargetTestPrep · Answer

Notice that n* (which is the same thing as n!) is divisible by all positive integers from 1 to n.

7* + 2 is divisible by 2 (since both 7* and 2 are divisible by 2)

7* + 3 is divisible by 3 (since both 7* and 3 are divisible by 3)

…

In general, 7* + k is divisible by k when k is between 2 and 7, inclusive (since 7* is divisible by k and k is divisible by k).

Therefore, none of the numbers between 7* + 2 and 7* + 7, inclusive, is a prime.

Answer: A

BrushMyQuant · Answer

Given that  n∗  denotes the product of all the integers from 1 to n, inclusive and we need to find how many prime numbers are there between  7∗  + 2 and  7∗  + 7, inclusive

7∗  = Product of all the integers from 1 to 7, inclusive = 1*2*3*4*5*6*7 = 7!

7∗  + 2 = 7! + 2 = 1*2*3*4*5*6*7 + 2 = 2*(1*3*4*5*6*7 + 1) = a Multiple of 2 =>   NOT a Prime Number  
Similarly, 7! is also a multiple of all numbers from 3 to 7
=> all of the numbers from 7! + 2 to 7! + 7 will be Non-Prime numbers.

So,  Answer will be A .
Hope it helps!

(Working below)

7∗  + 3 = 7! + 3 = 1*2* 3 *4*5*6*7 + 3 = a Multiple of 3 =>   NOT a Prime Number  
 7∗  + 4 = 7! + 4 = 1*2*3* 4 *5*6*7 + 4 = a Multiple of 4 =>   NOT a Prime Number  
 7∗  + 5 = 7! + 5 = 1*2*3*4* 5 *6*7 + 5 = a Multiple of 5 =>   NOT a Prime Number  
 7∗  + 6 = 7! + 6 = 1*2*3*4*5* 6 *7 + 6 = a Multiple of 6 =>   NOT a Prime Number  
 7∗  + 7 = 7! + 7 = 1*2*3*4*5*6* 7  + 7 = a Multiple of 7 =>   NOT a Prime Number

Watch the following video to learn the Basics of Functions and Custom Characters

https://www.youtube.com/watch?v=b7JoMX0gk8M

bumpbot · Answer

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For an integer n greater than 1, n* denotes the product

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