GMAT integers are an integral part of the GMAT Quantitative reasoning section. Candidates often wonder “What is an integer? Is zero an integer? What does it mean if integers are consecutive?” Basically,GMAT integers are all multiples of 1. They are all the positive whole numbers and their negative opposites, as well as zero. GMAT integer questions do not include fractions, percentages, or numbers with decimals (which rule out figures like pi). Integers are one of the key recurring elements on the GMAT Quantitative reasoning section. For beginners starting preparation for the GMAT quant section needs to know all the relevant rules and properties of integers, tips, and tricks for every kind of integer question, along with example questions.
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GMAT Integers Syllabus
The GMAT syllabus of integers are the multiples of 1 and these can be positive and negative values like -2 to 2. The excluded values are fractions, decimals, which also target pi.
Example: -3, -2, -1, 0, 1, 2, 3
Note:
- Integers can be infinite for both positive and consecutive negative
- 0 is also an integer despite it not having any unique features
Candidates also might get questions like:
- If n is a positive integer, then n(n+1)(n+2) is
- Is 0 considered divisible by any integer? For example, is 0 divisible by 3?
Is 0 an Integer GMAT? Is 0 even or odd GMAT?
Candidates might be wondering “Is zero even or odd GMAT?”. Yes, zero is an integer. It fits into many different GMAT arithmetics, and algebraic number systems like whole, natural, rational, and real numbers. Zero is also an integral part of the additive identity property, which states that the numeral sum of adding zero to any number is the number itself.
The number 0 is even. Zero, when divided by 2, has no remainder (and 2, 4, and so on). 0 is a special integer with its own set of properties.
Few More GMAT Integer Samples
Essential Properties of Integers
It is very important for the candidates to know the properties of integers. This will help the candidates get a better score in GMAT Score calculator. Here are a few important properties of integers GMAT along with GMAT properties of integers practice questions:
Even and Odd Integers
GMAT quant is almost certainly familiar with even and odd numbers. Still, candidates will need this baseline info to understand subsequent properties and GMAT Algebra. So candidates must make sure they know all of the following definitions before moving on.
| Properties | Explanation |
|---|---|
| Even | Any number that results in an integer when divided by 2 is an even integer. Even integers end in 0, 2, 4, 6, or 8. |
| Odd | Any integer that isn’t divisible by 2 (as in, doesn’t result in an integer when divided by 2) is an odd integer. Note that I didn’t say “any number”— 4.5 is not divisible by 2, but it’s also not an integer, so it’s not odd. Odd integers end in 1, 3, 5, 7, or 9. |
| Both | Non-integers can’t be even or odd. Only integers can be even or odd, because decimal places automatically rule out divisibility by 1 or by 2. So if you see a question on the GMAT that specifies that a certain number is even or odd, you know it must be an integer. |
GMAT Integer questions:
Adding and Subtracting
The rules for adding and subtracting with integers are fairly intuitive as well. However, candidates must make sure that they have reviewed all of them before moving on. Candidates can also follow the GMAT preparation Tips to Ace the GMAT score in all the sections.
Any integer plus or minus another integer results in an integer.
4+5=9
4–5=−1
| Properties | Definitions | Explanation |
|---|---|---|
| Negative and Positive | Negative and positive numbers have particular rules when it comes to addition and subtraction | Adding a negative integer is the same as subtracting the positive. 4+(−5)=4–5=−1 (−4)+(−5)=(−4)–5=−9 Subtracting a negative integer is the same as adding the positive (double negatives cancel out). 4–(−5)=4+5=9 (−3)–(−4)=(−3)+4=1 |
| Odds and Evens | Adding or subtracting two of the same kind results in an even integer; adding or subtracting odd and even results in an odd integer | Odd + or – odd = even 5+7=12 Even + or – even = even 12–8=4 Odd + or – even = odd 8+3=11 8–3=5 |
GMAT Integer questions:
Multiplying and Dividing
Multiplying and dividing with integers is right where things start to get a little more complex. It’s essential that candidates understand all of these moving parts before hitting the examples below.
First, the basics: multiplying integers with other integers always yields a result that is also an integer. However, dividing with integers isn’t so straightforward. 3 / 5, for example, would yield a GMAT fraction or a decimal, which is by definition not an integer. Same with 5 / 3.
| Properties | Definitions | Explanation |
|---|---|---|
| Negative and Positive | When multiplying or dividing numbers with the same sign, the result is always positive. When multiplying or dividing numbers with a different sign, the result is always negative. | Negative × or / negative = positive (−5)×(−5)=25 Negative × or / positive = negative (−5)×5=(−25) Positive × or / negative = negative 5×(−5)=(−25) Positive × or / positive = positive 5×5=25 |
| Odds and Evens | Any integer multiplied by an even number is even | Odd × odd = odd Even × even = even Odd × even = even |
| Multiples and Factors | A multiple is the product of an integer and another integer. In other words, an integer that is perfectly divisible by another integer (with nothing left over) is called a multiple of the latter integer. For example, 20 is a multiple of 4, because 20 is divisible by 4. | Together, we can simplify these rules into an always-true equation: Multiple/Factor =Integer |
| Quotient and Remainders | The remainder is what is leftover in a division problem. For factors and their multiples, the remainder will always be 0, because factors go into their multiples perfectly, without anything left over. The quotient is how many times an integer can fit into another integer, regardless of what’s leftover. For example, 20 divided by 3 is 6 with 2 leftovers, because 3×6=18. So 20 divided by 3 has a quotient of 6 and a remainder of 2. | If x and y are positive integers, the quotient and remainder (q and r, respectively) can be represented with these formulae: y=xq+r,and0≤r≤x For example, if y is 20 and x is 3: 20=(3)q+r 20=(3)(6)+2 20=18+2 0≤2≤3 q=6,r=2 So y is only evenly divisible by x if the remainder r=0. |
GMAT Integer questions:
Properties of the Integer 0
As indicated above, zero is an integer. But it does have its own special properties:
- 0 is an even integer (because 2 goes into it 0 times)!
- 0 is is the only integer that is neither positive nor negative. So if you are asked about “negative numbers,” this doesn’t include zero. But if you are asked about “non-positive” or “non-negative” numbers, this does include zero.
- Any number plus or minus 0 is that number:
n+0=n;n–0=n
- Any number multiplied by 0 is 0:
n×0=0
- You cannot divide by 0:
n/0
=undefined
1 has some special properties as well.
- 1 is an odd integer.
- Any number multiplied or divided by 1 is itself:
- Any number other than zero divided by itself is 1:
- The reciprocal of any number is 1 over that number. Any number other than zero multiplied by its reciprocal equals 1:
- Multiplying a number by -1 changes the sign but not the absolute value:
5×1=5; 5×(−1)=−5
Prime Numbers
As you likely already know, prime numbers are positive integers that can only be divided by themselves and 1.There is a predefined set of prime numbers. The first prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
GMAT Integer questions:
Prime Factorization
Every integer greater than 1 is either prime or can be broken down into prime factors. The expression of this is called a prime factorization.
As an example, here’s the prime factorization of 30:
30=5×6=2×3×5
GMAT Integer questions:
Exponents and Square Roots
For integer problems, there isn’t too much you need to know about square roots. Just make sure you understand the definition below.
If the square root of a given number is an integer, that means that that the number has to be an integer too and is thus a perfect square. The perfect squares are 1 (12), 4 (22), 9 (32), 16 (42), and so on.
These tiles are perfect squares—but not the kind that we’re talking about.
GMAT Integer questions:
Consecutive Integers on the GMAT
Consecutive integers GMAT problems come up again and again: you may also see the word “inclusive,” meaning that the set is inclusive of the first and last numbers (“all integers from 1 to n, inclusive” means that this set includes 1 and n).
{ -10, -9, -8, -7 }
The above is a set of consecutive integers or integers that follow each other in order. Consecutive integers can be represented GMAT algebraic expressions:
N+1,n+2,n+3…,wheren is an integer
| Properties | Explanation |
|---|---|
| Consecutive Even Integers | { 2, 4, 6, 8 } is a set of consecutive even integers. Consecutive even integers can be represented algebraically as well: 2n,2n+2,2n+4,2n+6… |
| Consecutive Odd Integers | { 1, 3, 5, 9, } is a set of consecutive odd integers. Here’s the algebraic representation of consecutive odd integers: 2n+1,2n+3,2n+5,2n+7… |
| Sum of Consecutive Integers and Divisibility | If the number of integers in a consecutive set is odd: the sum of all the integers is always divisible by that number. Let’s say c is the number of consecutive integers. In the set { 2, 3, 4, 5, 6 } we have c=5 consecutive integers. So the sum of 2 + 3 + 4 + 5 + 6 should be divisible by 5: 2+3+4+5+6=20 20/5 =4 |
GMAT Integer questions:
Factorials
Factorials have some properties that intersect with consecutive integer properties, and questions involving factorials also come up fairly frequently on the GMAT.
If n is an integer greater than 1, then n factorial, represented by the symbol n! is the product of all of the integers from 1 to n.
For Instance:
3!=1×2×3=6
5!=1×2×3×4×5=120
0! Is the same as1!=1
| Properties | Explanation |
|---|---|
| Factorials and Permutations | For example, the GMAT could have a question giving you 8 kinds of pasta, 3 kinds of sauce, and 2 kinds of vegetable mix-ins and asking you how many types of pasta you could make by combining one of each of these ingredients. In this case, the answer is simple multiplication: 8×3×2=48 |
| Factorials and Consecutive Integers | The product of n consecutive integers is always divisible by n! For example, let’s say we are given this set: { 3, 4, 5, 6 }. That set contains 4 consecutive integers or n=4, so the product of 3×4×5×6 is divisible= by 4!. 3×4×5×6 360 4×3×2×1=24 360/24 =15 |
GMAT Integer questions:
*The article might have information for the previous academic years, which will be updated soon subject to the notification issued by the University/College.
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