Factors, Multiples & Primes: The Building Blocks of Numbers

Numbers aren't just points on a line — they have an internal structure built from multiplication. Factors are the numbers that divide evenly into a number; multiples are what you reach by multiplying it; and primes are the indivisible building blocks that every other number is made from. This number sense underlies simplifying fractions, finding common denominators, and solving divisibility puzzles, so it's worth knowing cold.

This guide covers factors, multiples, primes, and the GCF and LCM, with worked and practice problems matched to real test difficulty at Northside Tutoring.

Why Number Sense Matters

Factors, multiples, and primes appear directly on the SSAT and support fraction and number-theory questions on every test. Finding the GCF helps you simplify fractions; the LCM gives you common denominators. Strong number sense makes a lot of arithmetic faster and more intuitive.

Factors vs. Multiples

A factor of a number divides into it with no remainder — the factors of 12 are 1, 2, 3, 4, 6, and 12. A multiple is the result of multiplying the number by a whole number — the multiples of 12 are 12, 24, 36, 48, … A handy way to keep them straight: factors are smaller (they fit inside), multiples are bigger (they grow outward).

Prime Numbers

A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11, 13, …). Note that 2 is the only even prime, and 1 is not prime (it has only one factor). Numbers with more than two factors are composite.

Prime Factorization

Every whole number greater than 1 breaks into a unique product of primes. Build a factor tree: 60 = 6 × 10 = (2 × 3) × (2 × 5) = 2² × 3 × 5. This prime "fingerprint" is the key to finding the GCF and LCM.

Greatest Common Factor

The GCF of two numbers is the largest factor they share. Find it from their prime factorizations by taking the primes they have in common (to the lowest power). For 12 = 2²·3 and 18 = 2·3², the shared primes are 2 and 3, so GCF = 2 × 3 = 6. The GCF is what you divide out to simplify a fraction.

Least Common Multiple

The LCM is the smallest number both divide into — useful for common denominators. From the prime factorizations, take each prime to its highest power. For 12 = 2²·3 and 18 = 2·3²: LCM = 2² × 3² = 36.

Where You'll See This — Test by Test

Factors, multiples, and primes are foundational across tests — tested directly on the SSAT and supporting fraction and number questions on the SAT and ACT.

Watch the Lesson

Sometimes a diagram needs a voice. In the short video below, one of our Northside tutors walks through the core idea and works through test-style problems in real time.

Worked Example Problems

These problems are calibrated to the difficulty you'll actually see on test day. Try each one before opening the solution.

Common Mistakes to Avoid

Practice Problems — You Try

Three problems below. Work each before checking the solution.

The Northside Method — How We Teach This 1-on-1

Reading a blog is a great starting point. But there's a meaningful gap between understanding a concept and reflexively applying it under timed conditions. That gap is exactly what our tutors close.

Every Northside student works through a four-step framework:

1. Assessment. We diagnose which specific skills are slowing your student down — not just whether they "get it" in the abstract.
2. Perfect-match coach. We pair them with an elite tutor (we accept only the top 1% of applicants) whose teaching style fits how your student actually learns.
3. Bespoke plan. A roadmap built around your student's target score, target timeline, and current pacing data.
4. Data-driven adjustment. Every session ends with a check on whether the student's accuracy and speed are moving in the right direction.

And if a student meets all eligibility requirements but doesn't hit the defined score improvement? We provide 5 additional hours of cohort learning at no cost. That's the Northside guarantee — built on 25 years of measured outcomes.