Integers
61 flashcards covering Integers for the GMAT Quantitative section.
Integers are the whole numbers you encounter in everyday math, including positive numbers like 5, negative numbers like -3, and zero. They don't include fractions or decimals, so 4.5 isn't an integer. Essentially, integers form the building blocks of arithmetic and algebra, helping you understand concepts like addition, subtraction, multiplication, and division without complications from non-whole values.
On the GMAT Quantitative section, integers show up in various question types, such as arithmetic problems involving operations, factors, multiples, and divisibility rules. You'll often see traps like mistaking negative integers for positive ones or overlooking zero's unique properties, which can lead to errors in equations or word problems. Focus on mastering integer properties, like even and odd rules, absolute values, and prime factorization, as these are key to solving problems efficiently and avoiding common pitfalls.
For a quick tip: Practice identifying factors and multiples to spot patterns faster.
Terms (61)
- 01
Integer
An integer is a whole number that can be positive, negative, or zero, such as -3, 0, or 5, and it has no fractional or decimal part.
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Even integer
An even integer is any integer that is divisible by 2 with no remainder, such as -4, 0, or 6.
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Odd integer
An odd integer is any integer that is not divisible by 2, leaving a remainder of 1 when divided by 2, such as -3, 1, or 7.
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Positive integer
A positive integer is an integer greater than zero, such as 1, 2, or 100.
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Negative integer
A negative integer is an integer less than zero, such as -1, -5, or -10.
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Zero
Zero is an integer that is neither positive nor negative and serves as the additive identity, meaning any integer added to zero remains the same.
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Absolute value of an integer
The absolute value of an integer is its distance from zero on the number line, so it is always non-negative, such as | -5 | = 5.
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Sum of two integers
The sum of two integers is the result of adding them together, and it can be even or odd depending on the parity of the integers involved.
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Difference of two integers
The difference of two integers is the result of subtracting one from the other, which is another integer.
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Product of two integers
The product of two integers is the result of multiplying them, and it is even if at least one factor is even.
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Quotient and remainder
When one integer is divided by another, the quotient is the integer result of the division, and the remainder is what is left over, as in 10 divided by 3 gives quotient 3 and remainder 1.
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Factors of an integer
Factors of an integer are the integers that divide it evenly without a remainder, such as 1, 2, 3, and 6 for the number 6.
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Multiples of an integer
Multiples of an integer are the results of multiplying it by any integer, such as 2, 4, 6, and so on for the number 2.
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Greatest common divisor (GCD)
The greatest common divisor of two integers is the largest positive integer that divides both without a remainder.
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Least common multiple (LCM)
The least common multiple of two integers is the smallest positive integer that is a multiple of both.
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Prime number
A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself, such as 2, 3, or 5.
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Composite number
A composite number is a positive integer greater than 1 that is not prime, meaning it has divisors other than 1 and itself, such as 4 or 6.
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Prime factorization
Prime factorization is the process of breaking down a positive integer into a product of its prime number factors, such as 12 = 2 × 2 × 3.
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Divisibility rule for 2
An integer is divisible by 2 if its last digit is even, such as 14 or 26.
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Divisibility rule for 3
An integer is divisible by 3 if the sum of its digits is divisible by 3, such as 12 because 1 + 2 = 3.
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Divisibility rule for 4
An integer is divisible by 4 if the number formed by its last two digits is divisible by 4, such as 124 because 24 is divisible by 4.
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Divisibility rule for 5
An integer is divisible by 5 if its last digit is 0 or 5, such as 15 or 20.
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Divisibility rule for 6
An integer is divisible by 6 if it is divisible by both 2 and 3, such as 18.
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Divisibility rule for 9
An integer is divisible by 9 if the sum of its digits is divisible by 9, such as 27 because 2 + 7 = 9.
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Divisibility rule for 10
An integer is divisible by 10 if its last digit is 0, such as 30 or 100.
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Consecutive integers
Consecutive integers are a sequence of integers that follow one another without gaps, such as 4, 5, and 6.
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Consecutive even integers
Consecutive even integers are even numbers that follow one another, such as 2, 4, and 6.
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Consecutive odd integers
Consecutive odd integers are odd numbers that follow one another, such as 1, 3, and 5.
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Sum of consecutive integers
The sum of a set of consecutive integers can be calculated using the formula for the sum of an arithmetic series, and it equals the number of terms times the average of the first and last term.
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Product of consecutive integers
The product of consecutive integers is always even if there are at least two integers, since one of them must be even.
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Even plus even
The sum of two even integers is always even.
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Even plus odd
The sum of an even integer and an odd integer is always odd.
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Odd plus odd
The sum of two odd integers is always even.
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Even times even
The product of two even integers is always even.
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Even times odd
The product of an even integer and an odd integer is always even.
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Odd times odd
The product of two odd integers is always odd.
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Remainder when dividing integers
The remainder is the amount left over after dividing one integer by another, and it is always less than the divisor.
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Modular arithmetic
Modular arithmetic deals with remainders when integers are divided by a fixed number, such as finding that 17 mod 5 equals 2.
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Properties of exponents with integers
When raising an integer base to an exponent, the result follows rules like multiplying exponents when raising a power to another power.
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Negative exponents
A negative exponent on an integer base means taking the reciprocal of the base raised to the positive exponent, such as 2^{-3} = 1/8.
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Zero exponent
Any non-zero integer raised to the power of zero equals 1.
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Order of operations with integers
When performing operations with integers, follow the standard order: parentheses, exponents, multiplication and division from left to right, then addition and subtraction.
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Integers in inequalities
Inequalities involving integers must consider the direction of the inequality when multiplying or dividing by negative numbers.
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Solving linear equations with integers
Linear equations with integer coefficients may have integer solutions, which can be found by isolating the variable.
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Trap: Even and odd in equations
A common error is assuming that if the product of two integers is even, both must be even, but only one needs to be even.
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Trap: Division by zero
Division by zero is undefined and can lead to errors in problems involving integers.
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Trap: Assuming all integers are positive
Problems may involve negative integers, so always check for signs in calculations.
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Example: Finding GCD of 12 and 18
The GCD of 12 and 18 is 6, as it is the largest number that divides both evenly.
Factors of 12 are 1, 2, 3, 4, 6, 12; factors of 18 are 1, 2, 3, 6, 9, 18; common are 1, 2, 3, 6.
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Example: LCM of 4 and 6
The LCM of 4 and 6 is 12, as it is the smallest number that is a multiple of both.
Multiples of 4: 4, 8, 12, 16...; multiples of 6: 6, 12, 18...; smallest common is 12.
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Example: Prime factors of 100
The prime factors of 100 are 2, 2, 5, and 5, written as 100 = 2^2 × 5^2.
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Example: Sum of first n integers
The sum of the first n positive integers is given by the formula n(n+1)/2, such as for n=5, sum is 15.
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Integers in word problems
In word problems, integers often represent quantities like ages or items, and must satisfy conditions such as being positive.
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Integers and fractions
Integers can be part of fractions, but when dealing with integer results, ensure the fraction simplifies to a whole number.
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Rounding to nearest integer
Rounding a number to the nearest integer means adjusting it to the closest whole number, such as 3.7 rounds to 4.
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Floor function
The floor function of an integer is the integer itself, but for non-integers, it gives the greatest integer less than or equal to the number.
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Ceiling function
The ceiling function of an integer is the integer itself, but for non-integers, it gives the smallest integer greater than or equal to the number.
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Absolute value inequality
An absolute value inequality like |x| < 5 means x is between -5 and 5, inclusive of integers in that range.
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Distance on number line
The distance between two integers on the number line is the absolute value of their difference, such as between -2 and 3 is 5.
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Ordering integers
Integers are ordered from least to greatest on the number line, with negative numbers before zero and positive after.
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Median of a set of integers
The median of a set of integers is the middle value when ordered, or the average of the two middle values if even-numbered.
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Mode of a set of integers
The mode of a set of integers is the value that appears most frequently, if one exists.