Prime Factorization on the SAT: Using Unique Factors to Solve Problems Efficiently

Prime factorization is expressing a number as a product of prime numbers only. For example, 24 = 2×2×2×3 = 2^3×3. This seems mechanical until you realize it makes finding GCF, LCM, and divisibility obvious. Instead of listing factors or testing divisibility, you see the prime factors and immediately understand relationships. Prime factorization transforms problems that feel complex into problems with transparent solutions. Students who master this skill solve several types of SAT problems faster than students who rely on other methods.

For example, finding GCF of 24 and 36 using prime factorization: 24=2^3×3, 36=2^2×3^2. The GCF is the common prime factors with the lowest powers: 2^2×3=12. This is instantly obvious from prime factorizations but requires listing factors without them. Finding LCM is similar: combine all prime factors with their highest powers: 2^3×3^2=72. These problems become almost trivial with prime factorization.

Use the division method: divide the number by the smallest prime (2) repeatedly until it is no longer divisible. Then divide by the next prime (3), then 5, 7, and so on. Track the prime factors as you divide. For example, 24: divide by 2 to get 12, divide by 2 to get 6, divide by 2 to get 3, divide by 3 to get 1. So 24=2^3×3. This systematic division is faster than any other method once you practice it, and it works for any number. On the SAT, you rarely need to prime-factorize numbers larger than 100, so practice speed with numbers in that range.

Shortcuts help: if a number is even, it has a factor of 2. If the digit sum is divisible by 3, the number is divisible by 3. If it ends in 0 or 5, it is divisible by 5. These checks help you divide faster. Over time, you become so practiced at prime factorization that you can prime-factorize most numbers under 100 mentally in seconds.

Once you have prime factorizations, finding GCF is mechanical: take common prime factors with their lowest powers. Finding LCM is mechanical: take all prime factors with their highest powers. This removes all guesswork. Instead of wondering whether you found the GCF correctly, you know you did because you applied the definition directly to the prime factorizations. This confidence and accuracy directly translate to more correct answers on the SAT.

Example: find GCF and LCM of 60 and 90. 60=2^2×3×5, 90=2×3^2×5. GCF: common factors with lowest powers = 2×3×5=30. LCM: all factors with highest powers = 2^2×3^2×5=180. Both are instantly obvious and verifiable. Without prime factorization, these are error-prone calculations. With it, they are certain.

Create a 10-minute daily drill: prime-factorize 5 random numbers between 10 and 100. Record time and accuracy. Within a week, you should factorize any number under 100 in under 15 seconds. This fluency makes GCF, LCM, and divisibility problems trivial because the hard part (finding prime factors) is automatic. Once it is automatic, you can focus on what to do with the factorization.

Practice GCF and LCM problems using prime factorization. After a week of fluency drills plus a week of application problems, these question types should feel easy. You will notice you solve them faster than classmates. This speed advantage comes from the systematic approach, not from superior math ability. Anyone who practices prime factorization builds this fluency. On test day, watch yourself solve these problems almost without thinking, knowing the answer is correct because you derived it from the prime factorizations systematically.