# Number Theory Vocabulary Puzzles

Our Number Theory Vocabulary Puzzles are a great way to hone students’ math vocabulary skills related to elementary math concepts. Our new puzzles do NOT require any Java applets. We have added new crosswords puzzles with three levels of difficulty, and our all-new word search. All resources are interactive, engaging, and include a timer. Solutions are also provided. Choose a puzzle below to get started. Be sure to try our related activities, too!

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Featured Number Theory Vocabulary Words

Congruence – In number theory, congruence refers to the relationship between two integers when their difference is divisible by a given integer, known as the modulus. For example, in modular arithmetic, if two numbers have the same remainder when divided by a modulus, they are said to be congruent.

Composite – In number theory, a composite number is a positive integer greater than 1 that has more than two distinct positive divisors. In other words, it is a non-prime number that can be factored into smaller positive integers other than 1 and itself.

Divisor – A divisor of an integer is a number that divides the integer without leaving a remainder. For example, the divisors of 12 are 1, 2, 3, 4, 6, and 12, as they can all divide 12 evenly.

Euclidean Algorithm – The Euclidean Algorithm is an efficient method for finding the greatest common divisor (GCD) of two integers. It works by repeatedly applying the property that the GCD of two numbers is the same as the GCD of one of the numbers and the remainder when the other number is divided by the first.

Factor – In number theory, a factor of a number is an integer that divides the number evenly without leaving a remainder. For example, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

Fermat’s Little Theorem – Fermat’s Little Theorem states that if p is a prime number and a is an integer not divisible by p, then
ap-1 ≡ 1(mod p). This theorem has applications in cryptography and number theory.

GCD (Greatest Common Divisor) – The greatest common divisor (GCD) of two integers is the largest positive integer that divides both of them without leaving a remainder. For example, the GCD of 12 and 18 is 6, as 6 is the largest number that divides both 12 and 18 evenly.

Integer – An integer is a whole number that can be positive, negative, or zero. It does not have a fractional or decimal part. Examples of integers include -3, 0, 7, and 101.

Modular Arithmetic – Modular arithmetic is a system of arithmetic for integers, where numbers “wrap around” after reaching a certain value called the modulus. It is often used in cryptography, computer science, and number theory.

Multiple – In number theory, a multiple of a number is the product of that number and an integer. For example, 12 is a multiple of 3 because 3 × 4 = 12.

Prime – A prime number is a positive integer greater than 1 that has exactly two distinct positive divisors – 1 and itself. For example, 2, 3, 5, 7, and 11 are prime numbers because they are only divisible by 1 and the number itself.

Prime Factorization – Prime factorization is the process of expressing a composite number as the product of prime numbers. It is a fundamental concept in number theory and is used in various mathematical computations, such as finding the GCD or LCM of numbers.

Primality Test – A primality test is an algorithm used to determine whether a given number is prime or composite. There are various primality tests, each with different levels of efficiency and accuracy.

Relatively Prime – Two integers are said to be relatively prime if their greatest common divisor (GCD) is 1. In other words, they have no common factors other than 1.

Residue – In modular arithmetic, the residue of an integer a modulo n is the remainder when a is divided by n. For example, the residue of 17 modulo 5 is 2, as 17 mod 5 = 2.

Sieve of Eratosthenes – The Sieve of Eratosthenes is an ancient algorithm used to generate all prime numbers up to a specified limit. It works by iteratively marking the multiples of each prime starting from 2.

Totient Function – The totient function, denoted by φ(n), is a function in number theory that counts the number of positive integers less than or equal to n that are relatively prime to n. It has applications in cryptography, particularly in RSA encryption.

Unit – In number theory, a unit is an element of a mathematical structure (such as a ring or field) that has a multiplicative inverse. In the context of integers, the only units are 1 and -1, as they are their own inverses.

Wilson’s Theorem – Wilson’s Theorem states that a positive integer n is prime if and only if (n−1)! ≡ -1(mod n). This theorem provides a necessary and sufficient condition for primality, though it is not typically used for primality testing due to its computational complexity.

Bezout’s Identity – Bezout’s Identity, also known as Bezout’s Lemma, states that for any two integers a and b, there exist integers x and y such that ax + by = gcd(a,b), where gcd(a, b) is the greatest common divisor of a and b.

Carmichael Number – A Carmichael number is a composite integer n that satisfies an – 1 ≡ 1(mod n) for all integers a relatively prime to n. Carmichael numbers are pseudoprimes to all bases and have applications in cryptography.

Divisibility – Divisibility is a property of integers where one integer can be divided by another without leaving a remainder. For example, 6 is divisible by 3 because 6 ÷ 3 = 2 with no remainder.

Legendre Symbol – The Legendre symbol, denoted as (a/p), is a mathematical concept in number theory used to determine whether a given integer is a quadratic residue modulo a prime number. It has applications in various areas, including cryptography and algebraic number theory.

Mersenne Prime – A Mersenne prime is a prime number that can be written in the form
2p – 1, where p is also a prime number. Mersenne primes are of particular interest due to their connection with perfect numbers and their use in cryptography.

Perfect Number – In number theory, a perfect number is a positive integer that is equal to the sum of its proper divisors, excluding itself. The smallest perfect number is 6, whose proper divisors are 1, 2, and 3, and 1 + 2 + 3 = 6.