A function g(n), where n is an integer, is defined as the product of

Question

A function g(n), where n is an integer, is defined as the product of all integers from 1 to n. How many of the followings must be a prime number?

g(11) + 5; g(11) + 6; g(11) + 7; and g(11) + 8?

A. 1 
B. 2 
C. 3 
D. 4 
E. None

CareerGeek · Accepted Answer

g(n) = 1*2*3.....*n = n!
--> g(11) = 11!

Note: Observe that g(11) = 11! can be written as 5A or 6B or 7C or 8D, where A,B,C & D are positive integers

g(11) + 5 = 5A + 5 = 5(A + 1) -->  Not Prime 
g(11) + 6 = 6B + 6 = 6(B + 1) -->  Not Prime 
g(11) + 7 = 7C + 7 = 7(C + 1) -->  Not Prime 
g(11) + 8 = 8D + 8 = 8(D + 1) -->  Not Prime

IMO Option E

Pls Hit Kudos if you like the solution

vishalsaraf · Answer

Is option 2 the answer because when you add an even number to an even number it will always be even and hence cannot be prime.

jfranciscocuencag · Answer

Hello  Dillesh4096

I know this are called co-primes and thats why they are not primes but, what happen if for example we have:

5! + 7 = 127 which is prime

How can we know if a number is prime or not, 5! is easy cuz we know thevalue but with bigger numbers?

Kind regards!

CareerGeek · Answer

Hi  jfranciscocuencag

In my opinion, GMAT would not ask questions like say g(6) + 7 which is 727. As finding the Prime nature of that number will be extremely difficult without any method.
But yes, i would expect them to ask the number you gave g(5) + 7 = 127.

However there is method to find whether a given number is prime or not

How to find a given number N is prime or not ?

Step 1: Find the nearest integral square root of N
Step 2: List all the prime numbers less than or equal to  \sqrt{N} 
Step 3: Find whether the given number N divides any of the numbers from step 2.

Conclusion: 1) If any number divides N, then N is NOT A PRIME Number
2) If no number divides N, then N is A PRIME Number

Eg1: Find whether 197 is prime or not ?

Step 1:  \sqrt{197}  --> Nearest integral square root = 14
Step 2: Prime numbers less than or equal to 14 --> 2, 3, 5, 7, 11, 13
Step 3: Divisibility: 2 - NO, 3 - NO, 5 - NO, 7 - NO, 11 - NO, 13 - NO.

Conclusion: 197 is A PRIME Number

Eg2: Find whether 459 is prime or not ?

Step 1:  \sqrt{459}  --> Nearest integral square root = 21
Step 2: Prime numbers less than or equal to 21 --> 2, 3, 5, 7, 11, 13, 17, 19
Step 3: Divisibility: 2 - NO, 3 -  YES , 5 - NO, 7 - NO, 11 - NO, 13 - NO, 17 -  YES , 19 - NO

Conclusion: 459 is NOT A PRIME Number

Though the above method would take around 2-2:30 minutes to solve. It's very unlikely that GMAT would test for such high numbers

Hope it's clear!

MagnusSamIBA · Answer

g(11) is 11!, which contains all primes up to 11.

Adding 5, 6, 7, or 8 to 11! doesn't make the result prime because these numbers retain divisibility by primes already present in 11!.

For example, g(11) + 5 is divisible by 5, g(11) + 6 is divisible by 2 and 3, g(11) + 7 is divisible by 7, and g(11) + 8 is divisible by 2 and 4.

Therefore, none of these expressions result in a prime number. The answer is E.

A function g(n), where n is an integer, is defined as the product of

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