 # Integer

### Why are GRE arithmetic questions on Integers important?

For every GRE aspirant, it is necessary that integers and its concepts are completely clear. One should ensure that the GRE arithmetic practice sets are solved as well. This is a prerequisite to excel in GRE since many aptitude questions use the terms such as; whole numbers, integers, and ratio, and familiarity with these words and concepts can prove to be highly beneficial.

### What are Integers?

Integers are whole numbers that can be represented on a number line and can be positive as well as negative. There is an equivalent negative number for each positive integer. Zero is a special type of integer, which neither belongs in the category of the positive nor negative integers. -10, -5, -3, 0, 2, 10 etc. are a few examples of integers. Any number represented on the left of zero on the number line is a negative integer, whereas, on right of zero is called as the positive integer.

### Types of Integer

When two or more integers can be written consecutively where a sequence is established based on the order of their size, then those integers are known as consecutive integers. So, an example from the GRE question paper based on integers will look like the below :

Calculate the number of consecutive integers, which are larger than -4 and smaller than 4?

This is a straightforward question based on the concepts of consecutive integers. Hence, the solution will be -3, -2, -1, 0, 1, 2, and 3. These seven numbers lie in between the range of -4 and 4, as their lower and upper bound.

### Absolute Value

Absolute value of any number is the distance of that number from the zero on the number line. The symbol used to represent the absolute value is | |. Since it is the distance; hence absolute value is always positive. Two numbers having same absolute value means that they are equivalent to one another on the number line; which means that these figures are the same, the only difference is, one number is positive, and the other is negative. For example the absolute value of -3 is |-3| = 3; meaning its distance is three units from the number zero.

### Other Properties of Integer

• Adding two integers: When two integers having the same sign are added, then add the absolute values of each integer keeping their original sign intact. For example; 8+2=10 and -10 + (-2)= -12
• Subtracting two integers: a – b = a + (−b) as well a – (−b)= a + b
• Multiplying two integers: (+)×(+)= +; (-)×(+)= -; (-)×(-)= +; (+)×(-)= –
• Dividing two integers: The quotient will be positive if the signs are the same otherwise it is negative.

even + even = even

odd + even = odd

odd + odd = even

odd × odd = odd

even × even = even

odd × even = even

Let’s solve a problem and get a proper insight of the type of questions that can be asked in GRE:

Question: What is the greatest integer, p, such that 5p is a factor of the product of the integers from 1 through 24, inclusive?

1. 1
2. 2
3. 3
4. 4
5. 5

Solution: For the purpose of the question, Let us express “the product of the integers from 1 through 24, inclusive”, as 24!

If 5p is a factor of 24, it should be evenly divided into 24, hence, for this to be true, p should not exceed the number of times 5 appears in the prime factorization of 24.

Hence the value of p will be the number of times 5 appears in the HCF of 24.

5 appear in: 5(1) =5, 5(2) = 10, 5(3) = 15, 5(4) = 20.

These are the only multiples of 5 that appear in the prime factorization of 24. So, there are four 5s in the prime factorization of 24. Hence, the value of p is 4. So, the correct answer is (d)

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