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Hi Sirs,
I have been having difficulty with DS questions involving number theory. I almost always tend to mark C as the answer even if the correct answer is A/B. Could you please suggest some strategy/tips to encounter such questions?

In general, there could be two possible reasons that might cause such difficulties with DS problems involving number theory:

1. The number theory itself (Make sure that you are familiar with all of its basic principles, and know how and when to apply them).
2. The format of the DS problems (One thing here is to work out an algorithm and always stick to it. For example, it may be helpful to write down the answer choices for each question and eliminate them as you analyze the statements. E.g. as soon as you understand, that the 1st statement is sufficient, cross out B, C and E).

To have a better understanding of what kind of difficulty you may have, could you please give a more specific example of a question that seems hard to you and try the following problems? (These are from GoGMAT platform, our students’ preference for diagnostic practice and improving particular weak points)

1. Is t a positive number?
(1) –t^2= t^3
(2) t^2 = 1

2. Is m-5 a prime number if m is a positive integer?
(1) 3m-1 is a prime number
(2) m-6 is an odd prime number

3. If c and d are positive integers, what is the largest prime factor of the product of 4c and 3d?
(1) The lowest common multiple of c and d is 90
(2) The greatest common divisor of c and d is 1.
Please, provide your explanation.

Hello sirs,
Sorry for the late response.

My answers to be three questions are -

1. A
t is positive as t^3 is positive (because -t^2 will always be positive)

2. B.
If m - 6 is an odd prime number, m - 5 will be even. So it cannot be prime.

3. C
If GCD of 2 nos is 1, then they are co-prime.
So LCM = product of nos = 90
Hence largest prime factor of 4c and 3d = largest prime factor of 90*4*3 = 5.

Great, thank you! A couple of comments to your solutions:

1. Expressions (-t)^2 and -t^2 are not equivalent: (-3)^2=3^2=9, but -3^2= - 9. I guess, here you considered the 1st statement as if it were (-t)^2 = t^3. Correct me, if I'm wrong;)
One more thing about your explanation: we cannot say, that (-t)^2 is always positive, as it can equal to 0, which is neither negative, nor positive.

Considering this information, will you still stick to answer A or will you change it?

2. Correct!

3. Here is the problem you wrote about: you tempt to answer C when one of the statements alone is sufficient.
I would recommend you to follow the simple algorithm:
1. Write down the question not to forget it:
2. Analyze the 1st statement ALONE and make your conclusion about its sufficiency
3. Analyze the 2nd statement ALONE as well
4. If both statements are not sufficient separately, analyze them together.

So, here we have:
1. what is the largest prime factor of the product of 4c and 3d? (Note, we are not asked about the value of the product, but only about its largest prime factor)

2. The lowest common multiple of c and d is 90
The LCM of the two numbers contains all the prime factors of these numbers, as it has to be divisible by both of them. So, LCM(a, b) and the product of a and b are not necessarily equal, but have he same prime factors. Hence, the greatest prime factor of 90 (5) is the greatest prime factor of the product of 4c and 3d.
The 1st statement alone is sufficient.

3. The greatest common divisor of c and d is 1.
The GCD tells us about common factors of c and d, but we do not know anything about the factors that are not in common. So the 2nd statement is not sufficient.

The answer here is A.

Try not to change the order of analysis of the statements even if the 2nd statement seems easier. And if you understand, that the two pieces of information together form some rule or formula, still check them alone.

Two problems below demonstrate some of the principles toched above. Please, let me know if there are any difficulties with them.
1. If x^n = l, where x and n are both integers, what is the value of the x?
(1) n is a multiple of 5.
(2) n is an odd number.
1. 2. If n and m are positive integers, what is the remainder when 3^(4n+5+m) is divided by 10 ?
(1) n = 2
(2) m = 1

Hello Sirs,

For the 1st question, yes i'd considered the 1st statement as if it were (-t)^2 = t^3.
Taking into consideration the possibility that (-t)^2 can equal zero,I think the answer should be E.


1. If x^n = l, where x and n are both integers, what is the value of the x?
(1) n is a multiple of 5.
(2) n is an odd number.
Step 1 - Considering statement 1.
n being a multiple of 5 can assume values 5, 10, 15, 20 ....
so x^5 = 1 => x = 1
but x^10 = 1 => x = +/-1
So, data is insufficient.
Step 2 - Considering statement 2.
n being odd can assume values 1,3,5,7...
so x^1 = 1 => x = 1
x^3 = 1 => x = 1
x^odd no = 1 => x = 1.
Hence statement 2 is sufficient to answer the question. Hence B.


2. If n and m are positive integers, what is the remainder when 3^(4n+5+m) is divided by 10 ?
(1) n = 2
(2) m = 1

The unit digits of powers of 3 (i.e remainder when divided by 10) has a cyclicity of 4. i.e 3, 9, 7, 1
so -
Step 1 - Considering statement 1.
n = 2. So 3^(8 + 5 + m). The remainder of this expression when divided by 10 cannot be determined. Hence data insufficient.
Step 2 - Considering statement 2.
m = 1. So 3^(4n + 6). (4n + 6)/4 will gives a remainder 2. Therefore (3^(4n + 6))/ 10 gives a remainder 9.
Hence B.


Please let me know if I m thinking on the correct lines. I m more concerned about my approach - which I now figure was majorly flawed.

Let’s first return to the problem

Is t a positive number?
(1) –t^2= t^3
(2) t^2 = 1

Though, your 1st explanation was not totally correct, you gave the right answer A. Indeed,
Statement (1): -t^2 is either negative, or 0, so t^3 is either negative, or 0 as well. Hence, t<=0. As 0 is neither positive, nor negative, we conclude, that t is a non-positive number. This means, that the answer to the question is “No”. The first statement is sufficient.
Statement (2): t = 1 or -1, so it is not sufficient. You were ok with this one.
Your answers and explanations to the last two problems are absolutely correct! And the approach is ok. My only comment to the 1st problem:
If x^n = l, where x and n are both integers, what is the value of the x?
(1) n is a multiple of 5.
(2) n is an odd number
Note, that from statement (1) n can equal 0, as 0 is a multiple of any integer except for itself. And if n=0, x can be actually any integer, except for 0. (5^0=1; (-5)^0=1 and so on, 0^0 is undefined). This fact about 0 can be crucial in some problems, still here your explanation is absolutely correct.
Keep on practicing this particular kind of problems (still don’t forget about all other types and principles), paying attention to details (such as properties of 0), following the strict algorithm in DS and always expecting some trick in each statement you consider.
If you have further difficulties, please, give examples of problems that seem hard and we’ll consider them together.

Hi Sirs,
Thanks a lot for your help with DS.
Another pain area for me is CR. Especially with 700+ level questions. I tend to narrow down to 2 answer choices and then almost always select the wrong one.
Any specific approach that you recommend here?

It is not quite clear what the problem is. Usually in such a situation people make mistakes because of 2 reasons:
1) they do not pay attention to the question type and its requirements
2) they do not notice some out of scope info or extreme language in the second answer.