Finding Roots Composite Numbers

STEP 1: Find all the factors of a number

Example: Say you want to find the factors of 8. The factors of 8 are all the numbers that will divide into 8 evenly. (In other words, they are not decimals like 2.38 or 4.1) So take the numbers from 1 to 8 and divide 8 by each of them. 1) 8 8) 8 2) 83) 84) 8 5) 86) 87) 8 What are factors and how do I find them? 8 42 decimal 1 If you get an answer with a decimal in it, the number you divided by is not a factor of 8, so cross out these answers. Now let’s look at the numbers that are left

1) 88) 82) 84) The numbers on top are the factors of 8, factors: 8, 4, 2 and 1 But, did you notice that the numbers on the top and the numbers you divided by (on the left) are the same? That’s because we are finding the factors two at a time, The number on the left and the number on top are both factors of 8. So to save time we don’t have to divide by every number from 1 to 8, we can go halfway and stop.

If we only have to find half of the factors, how do we know when we have gotten halfway and can stop? 1) Write the number with two little branches below it 8 2) Starting with ‘1 x 8’ Write all the pairs of factors that divide evenly into 8 1 * 8 2 * 4 4 * 2 8 * 1 3) This is where they start to repeat, STOP HERE! You don’t need to write these repeating numbers down If you write the factors of the number using the following system, you can see where your stopping point will be. All the factors of 8 are right here in this little box.

Practice: Find the factors of the following numbers * 12 2 * 6 3 * 4 1 * 32 2 * 16 4 * 8 1 * 81 3 * 27 9 * 9 1 * 48 2 * 24 3 * 16 4 * 12 6 * 8 7 * decimal Here’s where the numbers start to repeat 4 * 3, etc. so stop here. Factors of 12: 1, 2, 3, 4, 6, 12 5 * decimal 6 * decimal 7 * decimal 8 * 4(repeat) Make sure you check all the numbers up to the number on the bottom right, this is where they start to repeat. Factors of 32: 1, 2, 4, 8, 16, 32 Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 Factors of 81: 1, 3, 9, 27, 81 You can stop here since there are no more numbers between these two factors on the bottom We can stop checking numbers as soon as we reach this number Stop, since the next number is 8 Read the factors in this order Down the left side Up the right side

Step 2: Split the number into two factors 1 * 12 2 * 6 3 * 4 Use the splitting property to simplify the following: 1) Find all the pairs of factors-look for perfect squares 2) Find the pair with the largest perfect square 3) Write this pair in the following order: 4 is a perfect square 4) Take the square root of the perfect number Answer This is a square root, so look for perfect squares

Step 2: Split the number into two factors Use the splitting property to simplify the following: 1) Find all the pairs of factors-look for perfect squares 2) Find the pair with the largest perfect square 3) Write this pair in the following order: 4) Take the square root of the perfect number Answer This is a square root, so look for perfect squares 1 * 32 2 * 16 4 * 8 4 and 16 are perfect squares

1 * 81 3 * 27 9 * 9 Step 2: Split the number into two factors Use the splitting property to simplify the following: 1) Find all the pairs of factors-look for perfect squares This is a square root, so look for perfect squares 2) Double factors like this mean that the original number was a perfect square and this splitting process is unnecessary. 3) Take the square root of 81 (see perfect numbers chart) Note: Checking for Prime numbers should also be done before trying the splitting process because prime numbers cannot be broken up at all. Answer

Step 2: Split the number into two factors Use the splitting property to simplify the following: 1) Find all the pairs of factors-look for perfect squares 2) Find the pair with the largest perfect square 3) Write this pair in the following order: 4) Take the square root of the perfect number Answer This is a square root, so look for perfect squares 4 and 16 are perfect squares 1 * 48 2 * 24 3 * 16 4 * 12 6 * 8

Step 2: Split the number into two factors Use the splitting property to simplify the following: 1) Find all the pairs of factors-look for perfect cubes 2) Find the pair with the largest perfect cube 3) Write this pair in the following order: 4) Take the cube root of the perfect number Answer This is a cube root, so look for perfect cubes 27 is a perfect cube 1 * * 54 3 * 36 4 * 27 6 * 18 9 * 12

Practice Problems