# Properties of Integers

While all questions in the quantitative section of GMAT refer to Real Numbers by default, many questions further classify the number as being an Integer, or as belonging to a subset of integers. Hence knowing the properties of integers is an important part of your GMAT preparation.

The easiest way to understand an integer is as a number that does not require a decimal point.

Therefore, 25 is an integer. We can write it as 25.0 or 25.00, but that is not necessary.

On the other hand, 14.6 is not an integer. We cannot omit the decimal point.

There are three types of Integers:

Positive integers [1, 2, 3, 4 ….], Negative integers [-1, -2, -3, -4 ……] and Zero (0)

Remember, zero is neither positive nor negative.

The key terms we need to know when we work with integers and the properties associated with them are explained below.

- Classification of Integers

Whole Numbers

Whole numbers are a subset of Integers and include Zero and all positive integers

i.e. [0, 1, 2, 3, 4, 5, .....]

Natural Numbers

Natural numbers is another name for Positive Integers.

i.e. [1, 2, 3, 4, 5, .......]

Odd Numbers

Integers that are not divisible by 2 are called Odd Numbers.

They can be positive [1, 3, 5, 7 ….] or negative [-1, -3, -5, -7 ….]

Even Numbers

Integers that are divisible by 2 are called Even numbers.

They can be positive [2, 4, 6, 8 ….] or negative [-2, -4, -6, -8 ….].

Do remember that Zero [0] is also an even integer.

Prime Numbers

A Positive integer that is not divisible by any other integer except by 1 and itself is called a Prime Number. The smallest Prime Number is 2. It is also the only even prime number. One [1] is not a Prime Number.

- Properties of Odd and Even Numbers

On some questions, you may have to perform the key arithmetic operatons of addition, subtraction, multiplicaton etc. on integers and conclude whether the result is an odd integer or an even integer.

(i) Addition

Odd + Odd = Even

Even + Even = Even

Odd + Even = Odd

(ii) Subtraction

Odd - Odd = Even

Even - Even = Even

Odd - Even = Odd

Even - Odd = Odd

(iii) Multiplication

Odd × Odd = Odd

Even × Even = Even

Odd × Even = Even

(iv) Exponent

Odd ^ Odd = Odd

Even ^ Even = Even

Odd ^ Even = Odd

Even ^ Odd = Even

- Factors and Multiples of Integers

If an integer *‘x’* is divisible by another integer *‘y’*, then we say that *‘y’* is a factor of *‘x’*.

We can also say that *'x'* is a multiple of *'y'*

The numbers 1, 2, 5 and 10 all divide 10. Hence they are called factors of 10.

Likewise, we can say that 10 is a multiple of 1, 2, 5 as well as 10 itself.

Every integer greater than 1 will have at least two factors i.e. 1 and the number itself.

Prime numbers can have only two factors, 1 and the prime number itself.

For example, the factors of 7 are [1,7]

### Prime factors

When we express an integer as a product of prime numbers, then we are looking at the Prime factors of that integer.

Every integer can be expressed as a product of its prime factors in one and only one way.

For example,

12 = 6 × 2

But 6 is not a prime number. Hence we need to split further.

We get 12 = 6 × 2 = 2 × 3 × 2 = 2² × 3

2² × 3 is the prime factor equivalent of the number 12.

Similarly, 90 = 9 × 10

But neither 9 nor 10 is prime. So we split further.

90 = 9 × 10 = 3 × 3 × 2 × 5, which when regrouped can be written as 2 × 3² × 5

So, 2 × 3² × 5 is the prime factor equivalent of the number 90.

### Highest common factor (HCF / GCD)

When we have two or more positive integers, they may have various factors in common.

The highest among these common factors is called the Highest Common Factor (HCF) or the Greatest Common Divisor (GCD).

For example,

The numbers 24 and 30 have the following factors in common:

1, 2, 3, 6

Since the highest among them is 6, their HCF is 6

For two or more numbers, all common factors will be the factors of their HCF

For instance, in the above example we saw that the HCF of 24 and 30 is 6.

What about the other common factors? As we can see, they are nothing but factors of 6 [1, 2, 3]

This is always true. Therefore, if we know the HCF, we can work out all the other common factors as well.

When the HCF of any two numbers is one, they are said to be co-primes or relatively prime to each other.

For example, the HCF of 9 & 10 is 1, because the numbers do not share any common factors except 1.

Hence, 9 & 10 are called co-primes.

### Lowest common multiple (LCM)

When we have more than one number, they may have various multiples in common.

The lowest among these common multiples is called the Lowest Common Multiple or Least Common Multiple (LCM).

For example,

The numbers 6 and 8 have the following common multiples:

24, 48, 72, 96 …..

The smallest of these is 24. Therefore, their LCM is 24

For two or more numbers, all other common multiples will be multiples of their LCM

In the above example, 24 is the LCM of 6 & 8. But what is their next common multiple?

It is nothing but 24 × 2 = 48

And the next?

It is 24 × 3 = 72

## Some useful results for integers

Whenever we have 2 consecutive integers, then one of them will always be divisible by 2

Whenever we have 3 consecutive integers, then one of them will always be divisible by 3

Whenever we have 4 consecutive integers, then one of them will always be divisible by 4

and so on……

Whenever we have 2 consecutive even numbers, then one of them will be divisible by 2 while the other will be divisible by 4 as well.

When we form a 4-digit number such that the first two digits are the same as the last two digits (e.g. 2626, 4747 etc), then such a number will always be divisible by 101.

When we form a 6-digit number such that the first three digits are the same as the last three digits (e.g. 254254, 957957 etc), then such a number will always be divisible by 7, 11, 13 and 1001.

If we interchange the digits of a 2-digit number, then the positive difference between the original number and the new number will always be divisible by 9.

e.g.

24 becomes 42, and 42 – 24 = 18 which is divisible by 9

If we reverse the digits of a 3-digit number, then the positive difference between the original number and the new number will always be divisible by 9, 11 and 99.

e.g.

371 becomes 173, the difference 371-173 = 198, which is divisible by 99, and hence also by 9 and 11.