What is a prime factorization, and why write the factors in order?

Every whole number above 1 is a product of primes in exactly one way, apart from the order: 84 is 2 × 2 × 3 × 7. Writing the factors smallest first gives every problem a single answer to check against, and it makes repeated factors easy to count, which is what the GCD and LCM are built from.

How do you find the GCD and LCM of two numbers?

Factorize both. The greatest common divisor multiplies the primes they share, as many times as both have them; the least common multiple multiplies every prime either one has, as many times as the one with more. For 12 = 2 × 2 × 3 and 18 = 2 × 3 × 3 that gives a GCD of 6 and an LCM of 36.

Why does the sheet keep showing numbers like 51 and 91?

Those are the composites that look prime. A child who checks for 2 and 5 and stops will call 51 prime, but it is 3 × 17, and 91 is 7 × 13. About half of every primes page is composite, and most of those are odd numbers not ending in 5, so the only way through is to keep dividing.

Which range should I start with?

Within 100 suits grades 4 and 5, where the primes to try are 2, 3, 5 and 7. Within 500 needs 11, 13, 17 and 19 as well, and a factorization there can end with a two- or three-digit prime, so it fits grade 6 or a child who already factorizes quickly.

Factors & Primes Worksheets

Prime factorization, GCD and LCM, prime or composite · Grades 4–6

Three kinds of practice with factors and multiples on a notebook grid. Write a number as a product of primes, smallest first, with one box per digit of each factor; find the greatest common divisor and the least common multiple of a pair; or mark each number prime or composite. The primes practice leans on the numbers that fool children, odd composites such as 51, 57 and 91 that pass the checks for 2 and 5. Numbers go up to 100 or 500, and the multiplication sign prints as × or ·, following the convention of each language.

Skills: prime factorization, greatest common divisor, least common multiple, prime and composite numbers, divisibility. Format: printable, randomized on every load.

Sheets are randomized, free to print, and carry no answer key on purpose – why Super Awesome Math works this way.

How to do it

Prime factorization breaks a number into the primes that multiply to make it, and once two numbers are written that way their GCD and LCM can be read straight off.

Divide by the smallest prime that goes in exactly, starting at 2, then 3, then 5, and so on. For 60, 2 goes in: 60 = 2 × 30.
Keep dividing the result the same way until what is left is itself prime. 30 = 2 × 15, 15 = 3 × 5, so 60 = 2 × 2 × 3 × 5.
Write the factors smallest first. Every number has exactly one such list, which is what makes it useful.
For the greatest common divisor of two numbers, multiply the primes that appear in both, taking each as many times as it appears in both lists.
For the least common multiple, multiply every prime that appears in either list, taking each the greater of the two numbers of times.

Where it goes wrong

Stopping at a composite. 60 = 4 × 15 is a true factorization but not a prime one: 4 and 15 both break down further.
Treating 1 as prime. It is not, and including it changes nothing while making the list wrong.

Frequently asked questions

What is a prime factorization, and why write the factors in order?: Every whole number above 1 is a product of primes in exactly one way, apart from the order: 84 is 2 × 2 × 3 × 7. Writing the factors smallest first gives every problem a single answer to check against, and it makes repeated factors easy to count, which is what the GCD and LCM are built from.
How do you find the GCD and LCM of two numbers?: Factorize both. The greatest common divisor multiplies the primes they share, as many times as both have them; the least common multiple multiplies every prime either one has, as many times as the one with more. For 12 = 2 × 2 × 3 and 18 = 2 × 3 × 3 that gives a GCD of 6 and an LCM of 36.
Why does the sheet keep showing numbers like 51 and 91?: Those are the composites that look prime. A child who checks for 2 and 5 and stops will call 51 prime, but it is 3 × 17, and 91 is 7 × 13. About half of every primes page is composite, and most of those are odd numbers not ending in 5, so the only way through is to keep dividing.
Which range should I start with?: Within 100 suits grades 4 and 5, where the primes to try are 2, 3, 5 and 7. Within 500 needs 11, 13, 17 and 19 as well, and a factorization there can end with a two- or three-digit prime, so it fits grade 6 or a child who already factorizes quickly.

Settings

Practice: prime factors, GCD and LCM, or prime or composite
Range: numbers within 100 or within 500
Sign: × or ·
Columns: 2, 3 or 4, as many as the practice allows
Print an answer key

Example problems

84 = □ × □ × □ × □
90 = □ × □ × □ × □
51 → □
91 → □

More Number sense

Comparison – Greater than, less than, equal
Rounding – Round to nearest 10, 100, 1000
Squares & Square Roots – Squares, roots of perfect squares, simplifying radicals