Math is all about numbers and you can imagine that quite a few concepts fall under properties of numbers.  Practice is the only way of familiarizing yourselves with this type of question but there a few things that you can keep in mind as you work your way through Grockit.

Even and Odd Numbers

Sometimes, all that a question tells you is that the product of two distinct numbers x and y is not even.  It then wants to know if x – y is always even.  How do you figure this out?

One way is to plug in numbers.  Suppose x = 5 and y = 3.  The product of x and y is 15 (odd) and x – y is 2 which is even.  Try this with a few other numbers and you can safely guess that x – y is always even if xy is odd.

Another way is to remember that

ODD * ODD = ODD

EVEN * EVEN = EVEN

ODD * EVEN = EVEN

In addition, for any number n, 2n is always even and 2n+1 is always odd.  Try it.  If n = 4, 2n = 8 which is even and 2n+1 = 9 which is odd.  If n = 5, 2n = 10 which is even and 2n+1 = 11 which is odd.

Knowing this, you can use it to solve the above problem.  If xy is odd, that means x and y must both be odd.  That means that if x = 2n + 1 and y = 2m + 1, then x – y = 2n – 2m = 2(n – m).  2 times of anything is always even, and thus, you have just proved algebraically that x – y is always even.

Other good things to remember is that

EVEN + EVEN = EVEN

ODD + ODD = EVEN

ODD + EVEN = ODD

Estimating quickly

Being about to round up and down and estimate quickly is always a good skill.  Knowing how to estimate fractions, in particular, can save you a fair bit of time on the GMAT.

Given this question: which of the following fractions is greater than ¼?

1. 12/50
2. 3/11
3. 2/9
4. 4/17
5. 6/24

You should be able to recognize quickly that 6/24 = ¼ so choice E is out.  Next look at choices C and D.  9 is close to 8 and ¼ of 8 is 2.  That means 2/9 is less than 2/8 i.e. 2/9 is less than ¼.  The same can be said of 4/17.  17 is close to 16 and ¼ of 16 is 4.  That means that 4/17 < 4/16.

Looking at choice A, any fraction over 25 or 50 can easily be converted into its decimal form.  12/50 = 24/100 = .24 < .25  So choice A is out too.  That leaves us with Choice B as the answer

Prime Numbers

By definition, prime numbers are only divisible by 1 and themselves.  This is a very useful property to know.  For example, if x and y are distinct prime numbers, then you know that x/y is not an integer because y is not a factor of x, or x would not be prime.

Prime factors

Knowing how to prime factorize will help you tackle almost any factorization problem.  Take the number 70 for example.  In order to find the prime factors of 70, divide 70 by the smallest prime, 2.  If that’s not possible, move on to the next prime.  70 is divisible by 2, which gives you 35.

Try dividing 35 by 2 again.  If that’s not possible, more on to the next prime.  35 is not divisible by 3, so try 5.  35/5 = 7 and since you are left with a prime number itself, you can stop there.

Through this constant dividing process, we can determine that 70 = 2*3*5

Try and see if you can find the prime factors of 244.

(You should get 168 = 2*2*2*3*7)

Let’s try applying it to this problem: The product of three positive integers is 70. If all of the integers are greater than 1, what is the sum of the greatest two integers?

1. 2
2. 7
3. 12
4. 14
5. 35

We worked out just not that 70 = 2*5*7, meaning that the two largest integers are 5 and 7.  That means that the sum is 12.

Squares

The last important thing to know about numbers is what perfect squares are.  A perfect square is an integer that is the square of another integer.  For example, 4, 9, 16, 25, 36 are perfect squares since they are integers and they are the squares of 2, 3, 4, 5 and 6 respectively.

The following is a simple application of the above concept.  The only thing that might confuse you at first is the phrasing of the question.  Which of the following is NOT equal to an integer squared?

1. Root 16
2. Root 9
3. 27/3
4. 37-12
5. 49

The question is essentially asking, which of the 5 choices is not a perfect square.  After you simplify the choices, pick the one that cannot be square rooted.  (Did you get choice B?)