Prime numbers
52 flashcards covering Prime numbers for the GMAT Quantitative section.
Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. For instance, 2, 3, 5, and 7 are primes because they can't be evenly divided by any other numbers, while 4 is not since it's divisible by 2. They serve as the fundamental building blocks for understanding multiplication, factors, and divisibility in mathematics.
On the GMAT Quantitative section, prime numbers frequently appear in problems involving number properties, factors, multiples, and equations. You might encounter questions asking you to identify primes, perform prime factorization, or apply them in word problems related to greatest common divisors or least common multiples. Common traps include mistakenly treating 1 as a prime or forgetting that 2 is the only even prime, so focus on quick recognition and accurate factorization to avoid errors.
A concrete tip: Practice listing primes up to 100 to improve your speed and accuracy.
Terms (52)
- 01
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
- 02
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.
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is not a prime number
The number 1 is not considered a prime number because it does not have exactly two distinct positive divisors.
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Smallest prime number
The smallest prime number is 2, as it is the lowest natural number greater than 1 with no divisors other than 1 and itself.
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Only even prime number
is the only even prime number because all other even numbers greater than 2 are divisible by 2 and thus not prime.
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Prime factorization
Prime factorization is the process of breaking down a composite number into a product of prime numbers that multiply to the original number.
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Fundamental Theorem of Arithmetic
The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be uniquely expressed as a product of prime numbers, up to the order of factors.
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Divisibility rule for 2
A number is divisible by 2 if its last digit is even, which is useful for identifying factors when working with primes.
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Divisibility rule for 3
A number is divisible by 3 if the sum of its digits is divisible by 3, helping to check for prime factors quickly.
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Divisibility rule for 5
A number is divisible by 5 if it ends in 0 or 5, which aids in prime factorization problems.
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Divisibility rule for 7
To check if a number is divisible by 7, double the last digit and subtract it from the rest of the number; if the result is divisible by 7, so is the original number.
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Trial division for primality
Trial division is a method to determine if a number is prime by checking for divisors from 2 up to the square root of the number.
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Number of distinct prime factors
The number of distinct prime factors of a number is the count of different prime numbers in its prime factorization.
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Total number of factors from primes
If a number's prime factorization is p^a q^b r^c, the total number of factors is (a+1)(b+1)(c+1), which is key for counting problems.
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Greatest common divisor with primes
The greatest common divisor (GCD) of two numbers can be found using their prime factorizations by taking the product of the lowest powers of common prime factors.
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Least common multiple with primes
The least common multiple (LCM) of two numbers is calculated from their prime factorizations by taking the highest powers of all primes present.
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Prime factors in exponents
In expressions with exponents, prime factors help simplify or compare numbers, such as determining if one is a multiple of another.
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Simplifying fractions with primes
To simplify a fraction, divide the numerator and denominator by their common prime factors to reach the lowest terms.
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Common trap: 9 is prime
A common mistake is thinking 9 is prime, but it is not because it is divisible by 3, so always verify by checking divisors.
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Common trap: Negative primes
Negative numbers are not considered prime because primes are defined as positive integers greater than 1.
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Is 17 a prime number?
is a prime number because it has no divisors other than 1 and 17 when checked up to its square root.
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Prime factors of 12
The prime factors of 12 are 2, 2, and 3, as 12 can be factored into 2 × 2 × 3.
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Prime factors of 36
The prime factors of 36 are 2, 2, 3, and 3, since 36 = 2² × 3².
- 24
Is 91 a prime number?
is not a prime number because it is divisible by 7 and 13, as 7 × 13 = 91.
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Strategy for large prime checks
For large numbers, check divisibility only by primes up to the square root to efficiently determine if it is prime.
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Primes in arithmetic sequences
Prime numbers can appear in arithmetic sequences, like 3, 5, 7, but not all sequences contain primes beyond the first few terms.
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Co-prime numbers
Two numbers are co-prime if their greatest common divisor is 1, meaning they share no common prime factors.
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Relatively prime pairs
Relatively prime pairs are numbers that have no common prime factors, which is important in fraction simplification and probability.
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Sum of prime factors
The sum of prime factors of a number is the total of its prime factors counted with multiplicity, useful in certain algebraic problems.
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Product of primes in a range
The product of all primes up to a certain number can be calculated for problems involving multiples or divisors.
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Primes and modular arithmetic
In modular arithmetic, primes are used to check remainders, such as determining if a number is divisible by a prime modulo that prime.
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Probability of selecting a prime
In a set of numbers, the probability of selecting a prime is the number of primes divided by the total numbers, often tested in counting problems.
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Even sum of two primes
The sum of two prime numbers is even only if both are odd or one is 2, since odd + odd = even and 2 + even prime doesn't exist beyond 2.
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Odd primes
All prime numbers greater than 2 are odd, as any even number larger than 2 is divisible by 2 and thus not prime.
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Prime triplets
Prime triplets are sets of three prime numbers that differ by a small amount, like 3, 5, 7, and may appear in sequence problems.
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Twin primes
Twin primes are pairs of primes that differ by 2, such as 3 and 5, and can be relevant in pattern recognition on the test.
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Largest prime less than 100
The largest prime less than 100 is 97, which has no divisors other than 1 and itself.
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Primes less than 20
The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19, useful for quick reference in problems.
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Factors of a prime
A prime number has exactly two factors: 1 and itself, which distinguishes it from composite numbers.
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Multiples of a prime
Multiples of a prime are numbers that can be expressed as the prime times an integer, like 2, 4, 6 for prime 2.
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Prime in exponents and roots
When dealing with exponents or roots, primes help simplify expressions, such as factoring out prime squares under a root.
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Common trap: All factors are primes
Not all factors of a number are primes; for example, 12 has factors 1, 2, 3, 4, 6, and 12, where only 2 and 3 are primes.
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Prime factorization of 100
The prime factorization of 100 is 2 × 2 × 5 × 5, or 2² × 5².
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Is 2 a prime number?
Yes, 2 is a prime number because it is greater than 1 and has no divisors other than 1 and 2.
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Prime numbers in fractions
Prime numbers in the numerator and denominator of fractions can be canceled if they match, simplifying the fraction.
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Distinct prime factors of 30
The distinct prime factors of 30 are 2, 3, and 5, as 30 = 2 × 3 × 5.
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GCD of 12 and 18
The GCD of 12 and 18 is 6, found by taking common prime factors: 12 = 2² × 3, 18 = 2 × 3², so 2 × 3 = 6.
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LCM of 4 and 6
The LCM of 4 and 6 is 12, determined from prime factors: 4 = 2², 6 = 2 × 3, so highest powers give 2² × 3 = 12.
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Strategy for prime-related word problems
In word problems, identify key numbers and their prime factors to solve for divisors, multiples, or probabilities efficiently.
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Prime and composite in sets
In a set of numbers, distinguishing primes from composites helps in counting or selecting subsets for probability questions.
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Example: Sum of primes less than 10
The sum of primes less than 10 is 2 + 3 + 5 + 7 = 17, illustrating basic operations with primes.
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Common trap: 4 is prime
is not prime because it is divisible by 2, so always check for factors beyond 1 and the number itself.