For example, to factor the number 18, break it into 1 x 18, or into 2 x 9, or into 3 x 6.
Example: In this guide, we’ll find the prime factorization of 24.
Example: If you don’t know any factors of 24, try dividing it by small prime numbers. Let’s divide by 2 to get 24 = 2 x 12. We’re not done yet, but this is a good start. Since 2 is prime, this is an easy way to start when factoring any even number.
Example: 24 /\ 2 12
Example: 12 is not a prime, so we factor it again. Let’s use 12 = 2 x 6 and add it to the factor tree: 24 /\ 2 12 /\ 2 x 6
Example: 2 is a prime number. Bring the 2 from the second line down to the third. 24 /\ 2 12 / /\ 2 2 6
Example: 6 is a non-prime number and needs to be factored again. 2 is a prime number, so we just bring the 2s down to the next row. 24 /\ 2 12 / /\ 2 2 6 / / /\ 2 2 2 3
Check your work by multiplying the last line together. It should equal the original number. Example: The final line of our factor tree has nothing but 2s and 3s. These are both primes, so we’re finished. We can write the prime factorization of 24 as 24 = 2 x 2 x 2 x 3. The order of the factors does not matter. 2 x 3 x 2 x 2 is also a correct answer.
Example: In the factorization 2 x 2 x 2 x 3, how many times does 2 appear? Since the answer is “three,” we can simplify 2 x 2 x 2 with 23. The simplified prime factorization is 23 x 3.
Find the prime factorizations of the two numbers. The prime factorization of 30 is 2 x 3 x 5. The prime factorization of 36 is 2 x 2 x 3 x 3. Find a number that appears on both prime factorizations. Cross it out once on each list and write it on a new line. For example, 2 is on both lists, so we write 2 on a new line. We’re left with 30 = 2 x 3 x 5 and 36 = 2 x 2 x 3 x 3. Repeat until there are no more factors in common. There’s also a 3 on both lists, so write it on your new line to make 2 and 3. Compare 30 = 2 x 3 x 5 and 36 = 2 x 2 x 3 x 3. There are no more numbers left in common. To find the GCF, multiply all the shared factors together. We just have 2 and 3 in our example, so the GCF is 2 x 3 = 6. This is the largest number that is both a factor of 30, and a factor of 36.
For example, simplify the fraction 30/36. We already found out that the GCF is 6, so divide both the numerator and denominator by 6: 30 ÷ 6 = 5 36 ÷ 6 = 6 30/36 = 5/6
Start with two prime factorizations. For example, the prime factorization of 126 is 2 x 3 x 3 x 7. The prime factorization of 84 is 2 x 2 x 3 x 7. For each unique factor, compare the number of times it appears in each list. Pick a list where it appears the greatest number of times, and circle each instance. For example, 2 appears once in the factors of 126, but twice in the list for 84. Circle the 2 x 2 in the second list. Repeat for each unique factor. For example, 3 appears most often in the first list, so circle the 3 x 3 there. 7 only appears once on each list, so circle a single 7. (It doesn’t matter which list you choose when there’s a tie. ) Multiply all of your circled numbers together to find the LCM. In our example, the least common multiple of 126 and 84 is 2 x 2 x 3 x 3 x 7 = 252. This is the smallest number that has both 126 and 84 as factors.
For example, we want to solve 1/6 + 4/21. Using the method above, we can find the LCM of 6 and 21. The answer is 42. Turn 1/6 into a fraction with 42 as the denominator. To do this, solve 42 ÷ 6 = 7. Multiply 1/6 x 7/7 = 7/42. To turn 4/21 into a fraction with 42 as the denominator, solve 42 ÷ 21 = 2. Multiply 4/21 x 2/2 = 8/42. Now that we have the fractions in forms with the same denominator, we can add them together easily: 7/42 + 8/42 = 15/42.
