If N is a two-digit even integer

[amanda.m.fox1]

If N is a two-digit even integer, is N<20?
1. The product of the digits of N is less than the sum of the digits of N.
2. The product of the digits of N is positive.

I understand why neither Statement 1 nor Statement 2 is sufficient on its own. I also understand the explanation the MGMAT gives for the answer to this problem. I am confused, though, why we do not have to take negative answers into consideration here when the problem does not state that N > 0 at any point.

Thank you!!

[RonPurewal]

you are correct; the problem statement should say 'positive'.

BUT you can rest assured that, if you are considering DIGITS, PRIMES, or FACTORS, then you will •••NEVER••• have to think about negative values.
that would be a 'trick', and the GMAT contains zero tricks, ever.

[RonPurewal]

more about this here:
https://www.manhattanprep.com/gmat/foru ... ml#p116154

[RonPurewal]

the problem has been fixed. thanks again for pointing out the issue.

[EdwardL946]

In the updated wording of this problem, all is OK with me, except for the reason that 22 is not an acceptable number that would make the answer E (Neither I nor II are sufficient) and not C (I and II together are sufficient). If anyone can please provide greater explanation for the section below in bold that would be much appreciated!!

If N is a two-digit positive even integer, is N < 20?

(1) The product of the digits of N is less than the sum of the digits of N.

(2) The product of the digits of N is positive.


Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

N is a two-digit even integer. The tens digit can be anything except for 0. The units digit has to be 0, 2, 4, 6, or 8.

(1) NOT SUFFICIENT: Consider the values of N that can satisfy this statement. If the units digit is zero, then the product will always be zero; this will always be less than the sum of the digits.

For example, N could be 10, in which case the product of the digits is less than the sum of the digits. In this case, the answer to the question “is N < 20?” is yes. Alternatively, N could be 20, in which case the product of the digits is less than the sum of the digits. In this case, the answer to the question “is N < 20?” is no.

(2) NOT SUFFICIENT: If the product is positive, then the units digit can't be zero. In this case, N could be 12 and the answer to the question is yes. Alternatively, N could be 22 and the answer to the question is no.

(1) AND (2) SUFFICIENT: Statement 2 indicates that the units digit cannot be 0. All of the cases in statement 1, then, where the units digit is 0 must now be disregarded.

N could still be 12. The product, 2, is smaller than the sum, 3. In this case, the answer to the question is yes.

Could N be 22 or greater? If N is 22, then the product, 4, is equal to the sum, 4. You can’t choose N = 22, then (remember that only numbers that make both statements true can be chosen). Are there any other possibilities greater than 22 that are valid?

If N = 24, then the product is 8 and the sum is 6; this doesn't fit statement 1. If N = 26, then the product is 12 and the sum is only 8; this also doesn't fit statement 1 (and is, in fact, even further away from what you want!). If N = 32, then the product is 6 and the sum is only 5; if N = 34, then the product is 12 and the sum is only 7. The higher you go, the larger the product gets compared to the sum. In other words, no larger number will have a product that is smaller than the sum.

Since N also cannot be 20 or 22, it must be less than 20. The two statements together are sufficient.

The correct answer is C.

[Sage Pearce-Higgins]

Thanks for being clear with your question. When you're testing statements (1) and (2) together to decide if the answer is C or E, remember that you're looking for 2 cases that agree with the statements (and other data in the question stem) and that give different answers to the question. The case of N=22 is ruled out because you're told from statement (1) that "The product of the digits of N is less than the sum of the digits of N." In the case of N=22, the product of the digits is 4, which is not less than the sum of the digits (also 4). Therefore this case doesn't agree with statement (1) and you can't use it.

[EdwardL946]

Thank you for your help with this one! Very good point that it does not satisfy the initial requirements.

[Sage Pearce-Higgins]

You're welcome.