When a natural number is expressed as a product of two other natural numbers, those other numbers are factors of the original number. For example, two factors of 12 are 3 and 4, because 3 • 4 = 12.
When one number can be divided by another number with no remainder, we say the first number is divisible by the other number. For example, 20 is divisible by 4 (). If a number is divisible by another number, it is also a multiple of that number. For example, 20 is divisible by 4, so 20 is a multiple of 4.
A Prime Number can be divided evenly only by 1 or itself.
And it must be a whole number greater than 1.
Example: 7 can only be divided evenly by 1 or 7, so it is a prime number.
But 6 can be divided evenly by 1, 2, 3 and 6 so it is NOT a prime number (it is a composite number).
Let me explain ...
Some whole numbers can be divided up evenly, and some can't!
Example:
6 can be divided evenly by 2, or by 3:
6 = 2 × 3
Like this:
ordivided into 2 groups divided into 3 groups
But 7 cannot be divided up evenly:
And we give them names:
When a number can be divided up evenly it is a Composite NumberWhen a number can not be divided up evenly it is a Prime Number
So 6 is Composite, but 7 is Prime.
Many mathematical operations have an inverse, or opposite, operation. Subtraction is the opposite of addition, division is the inverse of multiplication, and so on. Squaring, which we learned about in a previous lesson (exponents), has an inverse too, called "finding the square root." Remember, the square of a number is that number times itself. The perfect squares are the squares of the whole numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 …
The square root of a number, n, written
is the number that gives n when multiplied by itself. For example,
because 10 x 10 = 100
Examples
Here are the square roots of all the perfect squares from 1 to 100.
Finding square roots of of numbers that aren't perfect squares without a calculator
1. Estimate - first, get as close as you can by finding two perfect square roots your number is between.
2. Divide - divide your number by one of those square roots.
3. Average - take the average of the result of step 2 and the root.
4. Use the result of step 3 to repeat steps 2 and 3 until you have a number that is accurate enough for you.
Example: Calculate the square root of 10 () to 2 decimal places.
1. Find the two perfect square numbers it lies between.
Solution:
32 = 9 and 42 = 16, so lies between 3 and 4.
2. Divide 10 by 3. 10/3 = 3.33 (you can round off your answer)
3. Average 3.33 and 3. (3.33 + 3)/2 = 3.1667
Repeat step 2: 10/3.1667 = 3.1579
Repeat step 3: Average 3.1579 and 3.1667. (3.1579 + 3.1667)/2 = 3.1623
Try the answer --> Is 3.1623 squared equal to 10? 3.1623 x 3.1623 = 10.0001
If this is accurate enough for you, you can stop! Otherwise, you can repeat steps 2 and 3.
Note: There are a number of ways to calculate square roots without a calculator. This is only one of them.
