Number properties

sud21 wrote:Which of the following is NOT prime?

1,556,551

2,442,113

3,893,257

3,999,991

9,999,991

x² - y² = (x + y)(x - y).

Some integers can be expressed as the DIFFERENCE OF TWO SQUARES:
99 = 100 - 1 = 10² - 2² = (10 + 1)(10 - 1) = (11)(99).
9991 = 10000 - 9 = 100² - 3² = (100 + 3)(100 - 3) = (103)(97).

Test this line of reasoning in the answer choices.
Note the following:
If the question stem includes the phrase which of the following, the correct answer is likely to be D or E.

E: 9,999,991 = 10,000,000 - 9.
Doesn't work:
10,000,000 is not a perfect square

D: 3,999,991 = 4,000,000 - 9 = 2000² - 3² = (2000 + 3)(2000 - 3) = (2003)(1997).
Since 3,999,991 is divisible by 2003 and 1997, D is NOT prime.

The correct answer is D.

Hi Mitch,

I am unable to apply this kind of approach in exam. Do we have any other approach?

I was thinking of doing a Divisibility rule test of 2,3,5,7 for all these options.

apart from div by 77 I am able to do every div test i.e. of 2,3,5 in mind

will this approach work?

Thanks & Regards,
Vishal

vishalwin wrote:Hi Mitch,

I am unable to apply this kind of approach in exam. Do we have any other approach?

I was thinking of doing a Divisibility rule test of 2,3,5,7 for all these options.

apart from div by 77 I am able to do every div test i.e. of 2,3,5 in mind

will this approach work?

Thanks & Regards,
Vishal

When a question stem asks whether a large integer is prime, the easiest approach -- in most cases -- is to test for divisibility by small factors such as 2, 3, 5 and 9.
Here, the correct answer choice is not divisible by any of these small factors, so this approach is not viable.
3,999,991 has only 4 factors: 1, 1997, 2003, and 3,999,991.
The approach used in my solution above seems the fastest way to determine that 3,999,991 is not prime.
Official problems that test the same concept:
https://www.beatthegmat.com/og-ps-query-t276099.html
https://www.beatthegmat.com/another-toug ... 67922.html