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GMAT Problem Solving & Data Sufficiency Math: Number Properties

GMAT Problem Solving & Data Sufficiency Math: Number Properties

Number Properties is one of the foundational concepts of the GMAT quantitative reasoning section. Some of the concepts this article covers are very simple, and some people may even consider them obvious yet they aren’t so obvious to everyone. It is extremely crucial that you are up to date with these concepts if you are out to achieve your GMAT goals.

Numbers Defined

Integer– A numeric value with only zero after the decimal point. That means ⅔, 5.2, 10.3, and ½ are all non-integers. Simply put, if you see the term integer in the exam, you should know that there are no fractions allowed for that particular problem.

Divisor/factor– A number that divides into another number evenly. The term generally applies to integers but technically it could apply to fractions. For example, ¼ is a factor of ½, because ½ ÷ ¼ = 2. However in GMAT, 97 percent of the time, the term is going to apply to integers. In this article, therefore, we will only consider divisors and factors in relation to integers. For example, 4 is a divisor or factor of 12. It goes into 12 exactly 3 times.

Multiple– A number that multiplies from another number. For example, 24 is a multiple of 12. A simplistic way of thinking about this is that factors/divisors are less than or equal to the number you are testing, while multiples are greater than or equal to the number that you are testing.   

Largest Factor  = Smallest Multiple = The number itself. 

If the number = 12, the largest factor =12, and smallest multiple = 12.

Basic Functions

Terms related to basic functions. 

Sum – The result of an addition

Difference – The result of a subtraction 

Product – The result of a multiplication 

Quotient – the result of a division 

Evens and Odds

The concept of evens and odds is one the exam will leverage rather often.

  • Even integer – An integer that is divisible  by 2
  • Odd integer – An integer that is not divisible by 2

When zero is divided by 2 there is no remainder, so technically zero is even.

Computing Evens and Odds

There are a few basic rules worth committing to memory that will help you in problem-solving and Data Sufficiency questions in your GMAT exam. They include:

  • Only the sum or difference of an odd and an even number is odd. For example, \(2 + 3 = 5\) and \(3 + 4 = 7\) (Odd), \(3 – 2 = 1\) and \(7 – 4 = 3\) (Odd)
  • Sum or difference of two even values is even. For example, \(4 + 2 = 6\) and \(4 + 8 = 12\) (Even), \(8 – 4 = 4\) and \(10 – 8 = 2\) (Even)
  • Sum or difference of two odd values is even. For example, \(3 + 1 = 4\) and \(3 + 5 = 8\) (Even),  \(3 – 1 = 2\) and \(5 – 3 = 2\) (Even).
  • The product of two even numbers is even. e.g \(2 × 2 = 4, 4 × 4 = 16\).
  • The product of two odd numbers is odd. e.g \(1 × 3 = 3,  3 × 5 = 15\)
  • Product of an odd number and an even number is even. e.g \(2 × 3 = 6, 3 × 4 = 12\).
  • There are no consistent rules defined for quotients.

Properties of 1

  • The product of any value (whether an integer or a variable), and 1 is itself. i.e \(2 × 1 = 2, 5 × 1 = 5, n × 1 = n\). 
  • The quotient of any value other than zero divided by 1 is itself. i.e \(2 ÷ 1 = 2, 5 ÷ 1 = 5, n ÷ 1 = n\).
  • The quotient of any value other than zero divided by itself is 1. i.e \(12 ÷ 12 = 1, 3 ÷ 3 = 1, n ÷ n = n\).
  • The implicit exponent of any value is 1. i.e \(2 =2^1, 5=5^1, n= n^1\).

Properties of Zero

  • Zero is the only non-negative and non-positive value. It is what defines negative and positive; any number less than zero is negative and any number greater than zero is positive. 
  • Adding or subtracting zero from any number results in no change in value. 
  • The product of any value and zero is zero. \(2 × 0 = 0, 25 × 0 = 0, n × 0 = 0\).
  • The quotient of any value divided by zero is non-real. When you divide any value by zero,  whether it is a numeric value or a variable, the result is infinity (∞). It is a non-real value because it can’t be expressed on the number line. For the purposes of the GMAT exam, division by zero is undefined and actually never tested. 
  • Absolute value is the distance from zero on the number line. Absolute Value is always \(\geq0\). That is, \(l-4l = 4\) because -4 is 4 units from zero on the number line. The symbol for absolute is the vertical lines on both sides of -4.

Note: All GMAT math is real.

Real Numbers

Real numbers are any values that can be expressed on a number line.

Rational numbers are numbers that can be expressed as fractions such as ½, ⅓, ⅘, etc. We know for instance that ½ on the number line is halfway between 0 and 1. To place ⅓ on the number line we just have to divide the distance between 0 and 1 into three equal parts. Even if a number cannot be expressed as a terminating decimal it is still rational if it can be expressed as a fraction. 

Irrational numbers are numbers that cannot be expressed as fractions or terminating decimals. For example \(-½⫪\) or \(⫪√2\). Do not convert irrational numbers to decimals unless you have been asked to approximate them.

\(⫪ = 3.14, √2 = 1.4\) and \(√3 = 1.7\). You may need to commit to memory these three approximations. They could come in handy in geometry, especially when doing approximations.

To compare irrational values and rational values, and as such translate them on the number line, you will need to use what you know to approximate what you don’t know. For example, if you want to place \(√17\) on the number line, that will be difficult because \(√17\) is an irrational number. But 16, a perfect square is right next to 17. \(√16 = 4\). The \(√17\) will be just a little bit bigger than 4, so we can place \(√17\) between 4 and 5, very close to 4 on the number line.

Examples

Set up your scratchpad by listing the answer choices vertically A through E and label the list as what you seek. (Furthest from 0).

All our answer choices are in irrational radical notations. This is a clear indication that you should not translate your question into an integer or decimal format. Instead, you need to evaluate by getting out of the irrational format and using common values we can place on the number line.

Square all values to simply,

A. \(\begin{align*}2\sqrt10 &= (2\sqrt10)^2\\&=4 × 10 = 40\end{align*}\).

B. \(\begin{align*}3\sqrt8 &= (3\sqrt8)^2\\& = 9×8 = 72\end{align*}\)         72 is further than 40. Eliminate A.

C. \(\begin{align*}4\sqrt5 &= (4\sqrt5)^2\\&= 16 ×5 = 80\end{align*}\)         80 is further than 72. Eliminate B.

D. \(\begin{align*}5\sqrt3 &= (5\sqrt3)^2\\&= 25×3 = 75\end{align*}\)        80 is further than 75. Eliminate D.

E. \(\begin{align*}6\sqrt2 &= (6\sqrt2)^2\\& = 36×2 = 72\end{align*}\)        80 is further than 72. Eliminate E.

How would this Apply to a Data Sufficiency Problem?

Is \(y – z\) even?

Given that:    \(y + 1 < 0\)

                       \(z – 2 < 0\)

Solution

We know that: eve – even = even

                          even – odd = odd

                          odd – odd = even

We need to determine if y and z are both either even or odd because if they are different, the difference will be odd. 

i)  \(y + 1 < 0\) implies \(y < -1\). This statement by itself is not sufficient to make a determination.

ii)  \(z – 2 < 0\) implies \(z < +2\).  This statement by itself is not sufficient to make a determination.

Combining the two does not tell us whether the difference is even or odd. So even with both statements, we cannot make a determination. Therefore, we eliminate choices A, B, C, and D and we are left with E because at no point did we have enough information to determine whether the answer to that question,” Is y – z even?” is always a yes or a no.

As you get ready to take your GMAT exam, take some time and apply some of these concepts in your daily practice of data sufficiency and problem-solving questions. You can take advantage of any of our GMAT packages that offer lots of study resources in this area. If it’s going to improve your scores, then it’s worth every single penny.



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