SAT Number Theory Essentials: Divisibility Rules, Factors, and Prime Numbers

Prime numbers appear frequently on the SAT, and divisibility rules save time when factoring. A prime is divisible only by 1 and itself, while composite numbers have multiple factors. Memorizing divisibility rules for 2, 3, 5, and 10 lets you factor quickly without trial-and-error division. A number is divisible by 2 if its last digit is even, by 5 if it ends in 0 or 5, by 10 if it ends in 0, and by 3 if the sum of its digits is divisible by 3. These shortcuts apply to both small and large numbers, making them invaluable for efficient problem-solving.

The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Knowing these by heart helps you recognize when a number is prime and when it factors. For composite numbers, breaking them into prime factors reveals their structure, which is essential for GCD and LCM problems. Practice identifying primes within mixed sets and factoring composite numbers into prime components to build automaticity.

GCD (greatest common divisor) and LCM (least common multiple) problems test whether you can find shared factors and common multiples. The prime factorization method—breaking numbers into their prime components—is the fastest and most reliable approach for both GCD and LCM. To find the GCD, identify the highest power of each prime that divides both numbers, then multiply those prime powers. To find the LCM, take the highest power of each prime that appears in either factorization and multiply them together. These methods work for any pair of numbers and eliminate guessing.

Example: For GCD(60, 90), factor as 60=2^2×3×5 and 90=2×3^2×5. The GCD uses the lowest powers: 2^1×3^1×5^1=30. For LCM, use the highest powers: 2^2×3^2×5^1=180. This systematic approach is faster and less error-prone than listing multiples or factors manually, especially on problems with larger numbers or three-number sets.

Word problems often ask for the largest group size (GCD) or the earliest time something repeats (LCM). Recognizing these contexts lets you solve efficiently. When a problem asks for the largest equal groups, use GCD; when it asks for the first time something repeats simultaneously, use LCM. For instance, if 60 red balls and 90 blue balls must be divided into equal groups with no remainder, use GCD(60, 90)=30 to get 30 groups with 2 red and 3 blue each. If buses arrive every 12 and 18 minutes, LCM(12, 18)=36 tells you they arrive together every 36 minutes.

Build the habit of translating word-problem language into mathematical operations. Phrases like "largest possible," "no remainder," "equal groups," and "divided evenly" signal GCD. Phrases like "first time together," "least," "repeats," and "same time" signal LCM. Practicing these translations on five real word problems will make the distinction automatic during the test.

Build speed by calculating GCD and LCM without writing out full factorizations for simple numbers. For example, GCD(12, 8)=4 and LCM(12, 8)=24 should be instant. When in doubt or with larger numbers, always write out the prime factorization to avoid errors that cost valuable points. Also verify your answer: the GCD should divide both original numbers evenly, and the LCM should be divisible by both original numbers. These checks take three seconds and catch most calculation mistakes.

Common errors include confusing GCD and LCM (use the lowest powers for GCD, highest for LCM), forgetting to include all prime factors, and making arithmetic mistakes during factorization. Practice a series of 10 mixed GCD/LCM problems daily until you can solve them quickly and accurately without self-doubt or rechecking every step.