Number Theory Basics Cheat Sheet

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.

Understanding Number Theory Basics

Divisibility is about exact grouping. If 48 objects can be placed into 6 equal groups with none left over, then 6 divides 48. This simple idea gives factors a physical meaning.

Factors usually come in pairs. For 36, a factor pair is 4 and 9 because their product is 36. Listing pairs carefully prevents missed factors and makes it easier to tell whether a number is prime.

Stop checking once the smaller factor has passed the square root of the number. Any later pair would repeat one already found. This shortcut becomes important when numbers get larger.

Prime factorization works because primes are the basic pieces from which positive whole numbers are built. It helps compare numbers without relying on a long list of factors. For example, shared prime pieces show what two quantities can be split into evenly.

This is useful when reducing fractions. To simplify 42 over 56, find the common prime pieces in the top and bottom, then cancel them through division. The same thinking handles repeating schedules.

If one machine is checked every 12 days and another every 18 days, their next shared check day is found by including every prime piece needed by either interval. Keep repeated primes visible. Treating 12 as just 2 times 3 loses one factor of 2 and leads to wrong results.

The Euclidean algorithm is often faster than factor trees for large numbers. Divide the larger number by the smaller one, then replace the original pair with the smaller number and the remainder. A common divisor of the first pair must divide the remainder, since the remainder is what is left after removing whole groups of the smaller number.

Repeating this process eventually produces a remainder of zero. The last nonzero remainder is the greatest common factor. This method matters in algebra because it gives a reliable procedure, not a guess based on spotting factors.

Write each division clearly and check that every remainder is smaller than the divisor. A remainder equal to or larger than the divisor signals an error in the division.

Modular arithmetic describes repeating cycles. A clock is a familiar example. After 12, the hour count returns to 1, so adding 15 hours has the same clock effect as adding 3 hours.

Calendar patterns, check digits, computer storage, and simple encryption all use this cycle idea. When working with a modulus, reduce numbers early by replacing them with their remainder. This keeps calculations small.

Still, be careful with subtraction because a negative result needs to be converted to an equivalent nonnegative remainder. For a cycle of 7, negative 2 has the same position as 5. Students should separate three ideas that are easy to mix up.

A factor divides a number exactly, a multiple is made by multiplying, and a remainder measures what cannot be grouped evenly. Precise vocabulary makes proofs and multi-step problems much easier to follow.

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