Number Theory Basics Cheat Sheet

A printable reference covering divisibility, primes, factors, multiples, GCF, LCM, modular arithmetic, and prime factorization for grades 8-12.

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Number theory is the study of whole numbers and the patterns they follow. This cheat sheet helps students recognize factors, multiples, primes, divisibility rules, and remainders quickly. These ideas support algebra, fractions, cryptography, contest math, and proof-based reasoning. A clear reference makes it easier to solve problems without memorizing every step separately. The most important tools are prime factorization, the greatest common factor, the least common multiple, and modular arithmetic. Prime factorization breaks a number into prime building blocks, such as $60 = 2^2 \cdot 3 \cdot 5$ . The GCF uses the smallest shared powers of primes, while the LCM uses the largest powers needed to include both numbers. Modular arithmetic focuses on remainders, using notation such as $a \equiv b \pmod{n}$ .

Key Facts

A positive integer greater than $1$ is prime if its only positive factors are $1$ and itself.
Every integer $n > 1$ has a unique prime factorization, such as $84 = 2^2 \cdot 3 \cdot 7$ .
For positive integers $a$ and $b$ , $\gcd(a,b) \cdot \operatorname{lcm}(a,b) = ab$ .
A number is divisible by $3$ if the sum of its digits is divisible by $3$ , and divisible by $9$ if the sum of its digits is divisible by $9$ .
A number is divisible by $2$ if its last digit is even, and divisible by $5$ if its last digit is $0$ or $5$ .
The Euclidean algorithm uses repeated division: if $a = bq + r$ , then $\gcd(a,b) = \gcd(b,r)$ .
The notation $a \equiv b \pmod{n}$ means $a$ and $b$ have the same remainder when divided by $n$ .
If $a$ divides $b$ , written $a \mid b$ , then there is an integer $k$ such that $b = ak$ .

Vocabulary

Prime number: A prime number is an integer greater than $1$ with exactly two positive factors, $1$ and itself.
Composite number: A composite number is an integer greater than $1$ that has more than two positive factors.
Factor: A factor of a number is an integer that divides the number evenly with remainder $0$ .
Greatest common factor: The greatest common factor, or $\gcd$ , is the largest positive integer that divides two or more numbers.
Least common multiple: The least common multiple, or $\operatorname{lcm}$ , is the smallest positive integer that is a multiple of two or more numbers.
Congruence: A congruence $a \equiv b \pmod{n}$ means $a$ and $b$ differ by a multiple of $n$ .

Common Mistakes to Avoid

Calling $1$ a prime number is wrong because prime numbers must have exactly two positive factors, and $1$ has only one positive factor.
Finding the LCM by multiplying numbers every time is inefficient and can be wrong if the numbers share factors, since $\operatorname{lcm}(a,b)$ is not always $ab$ .
Confusing GCF and LCM leads to reversed answers because the GCF is the largest shared factor, while the LCM is the smallest shared multiple.
Using divisibility by $3$ or $9$ on the last digit is wrong because those tests depend on the sum of all digits, not the final digit.
Writing $a \equiv b \pmod{n}$ when the remainders are different is incorrect because congruent numbers must differ by a multiple of $n$ .

Practice Questions

1 Find the prime factorization of $360$ .
2 Find $\gcd(48,180)$ and $\operatorname{lcm}(48,180)$ .
3 Determine whether $157 \equiv 37 \pmod{12}$ , and explain using remainders.
4 Explain why prime factorization makes it possible to find both the GCF and the LCM of two numbers.

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