If n is a positive integer greater than 16, is n a prime

Question

If n is a positive integer greater than 16, is n a prime number?

(1) n is odd
(2) The remainder when n is divided by 3 is 1, and the remainder when n is divided by 7 is 1.

Bunuel · Answer

If n is a positive integer greater than 16, is n a prime number?

(1) n is odd. An odd number can be prime as well as non-prime. Not sufficient.

(2) The remainder when n is divided by 3 is 1, and the remainder when n is divided by 7 is 1:
\(n=3q+1\), so \(n\) can be; (1, 4, 7, 10, 13, 16), 19, 22, ... (numbers greater than 16)
\(n=7p+1\), so \(n\) can be; (1, 8, 15), 22, 29...

General formula of \(n\) based on the above two conditions would be \(n=21t+22\), divisor will be the least common multiple of above two divisors 3 and 7, hence 21 and the remainder will be the first common integer in above two patterns, hence 22 (for more on this check: manhattan-remainder-problem-93752.html?hilit=derive#p721341 and when-positive-integer-n-is-divided-by-5-the-remainder-is-90442.html?hilit=derive#p722552).

Next, from \(n=21t+22\) \(n\) can be: 22, 43, 64, 85, 106, 127, ... We can see that \(n\) ca be prime (for example 43) as well as non-prime (for example 85). Not sufficient.

(1)+(2) \(n\) can still be a prime as well as non-prime, for example 43 or 85. Not sufficient.

Answer: E.

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TGC · Answer

n>0 (Integer)
n> 16

Is n prime?

(1).

n=odd so n can be 17,19,21 .... 
Here 17 is prime. However, 21 is not. And hence INSUFFICIENT.

(2).

n = 3A +1 19,22,25,28,31,34,37,40, 43 
n= 7B + 1 22,29,36, 43

Combining series n = 21A + 43

So Numbers 43,64,85
Here 43 is prime . However, 64 isnt Hence INSUFFICIENT

Combining (1) and (2).

43,85.............While 85 isn't prime 43 is .

Insufficient Hence (E) it is

gmatchile · Answer

They indicate that n is greater than 16 and asked if n is Prime?

With condition 1) N is odd, there are many odd older than 16 who are not prime numbers, example numbers: 21, 25 and more. Eye and there are also many odd greater than 16, that if are prime numbers for example: 17, 19, and more.

Given the situation that there is both Prime and non-prime numbers, we conclude that the condition 1) is not enough.

Condition 2) rest 1, is obtained by dividing N by 3, and rest 1 is obtained by dividing N by 7.
This results in N = 3 K + 1 and N = 7 p + 1, given that N is unique and that 3 and 7 are prime numbers, we can conclude that N to be divided by (7 x 3) 21, also gets rest 1, since N must be divisible simultaneously for 3 and 7 for rest 1.

So N = 21J + 1, then N may be 22, 43, and other values, given that 22 is even and 43 is prime number, there is ambiguity, then 2) is not enough.

Let’s see if it is enough 1) and 2).

1) N = 17, 19, 21, 25,..., 43,..., 85,...

(2) N = 22, 43, 64, 85...

There we can see two values that satisfy both conditions 43 and 85, and as 43 is a prime number and 85 it is not a prime number, again produced ambiguity and we conclude that the two conditions together nor we can affirm or deny, initial sentence.

Conclusion the answer is E.

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If n is a positive integer greater than 16, is n a prime

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GMAT Data Sufficiency (DS) Questions

If n is a positive integer greater than 16, is n a prime

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