Multiplication and division can bring certain concepts that kids often get confused with; these concepts are factors, multiples and powers. These three terms we often use when we need to do multiplication or division in arithmetics. In this article, we shall try to make a distinction between these terms and the concepts related to them.
What are the Factors?
Factors of a number are simply the numbers which can divide the given number completely without leaving a remainder. Let us understand this with an example. Suppose we have to find factors of 24. Let’s list all those numbers, which can divide into 24.
24 ÷ 1 = 24
24 ÷ 2 = 12
24 ÷ 3 = 8
24 ÷ 4 = 6
24 ÷ 6 = 4
24 ÷ 24 = 1
Now, we see that 1, 2, 3, 4, 6, and 24 are the numbers which can divide 24 without leaving any remainder. Hence, factors of 24 are 1, 2, 3, 4, 6 and 24.
|Some important facts to remember.1 is a factor of every number.Every number is a factor of itself.Any number with only two factors — 1 and itself is called a prime number.Any number with more than two factors is called a composite number.1 has no factors other than itself.Prime factors — factors of a number which are prime.|
What are Multiples?
We all learn multiplication tables of numbers; the numbers we get in the times table of any number are called multiples of that number. For example, 2, 4, 6, 8, …, are the multiples of 2 and 3, 6, 9, 12, 15, …, are the multiples of 3.
Learn more about multiples of a number here.
|Some important facts to remember.There are infinitely many multiples of a number.The nth multiple of a number is determined by multiplying that number by n. For example, the 12th multiple of 4 is 4 × 12 = 48.The number itself is a factor of each of its multiple. For example, 12 is a multiple of 6, whereas 6 is a factor of 12.Multiples of a number make a number pattern. For example, take multiples of 5 — 5, 10, 15, 20, 25, 30, …; they form a number pattern where each term differs by 5.|
What is Meant by Power?
To add a number repeatedly, we can alternatively represent it as 2 + 2 + 2 + 2 = 4 × 2. But if we have to multiply a number repeatedly, we represent it as 2 × 2 × 2 × 2 = 24. The number 24 is called an exponent, where 2 is its base and 4 is its power. We say 24 as “2 raised to the power 4”. Hence, the number of times a number multiplied by itself is called its power.
- If a number p is multiplied twice by itself — p2, we say it is “p squared”.
- If a number p is multiplied thrice by itself — p3, we say it is “p cubed”.
- If a number p is multiplied n times by itself (n ≠ 2, n ≠ 3) — pn, we say it “p raised to the power n”.
Read more about exponent here.
There are certain laws of exponents that we use while dealing with power numbers.
- Multiplication of exponents whose bases are the same but powers are different.
|am × an = a m + n|
- Division of exponents whose bases are the same but powers are different.
|am ÷ an = a m – n|
- Negative powers
|a – m = 1/am|
- Zero power
|a 0 = 1|
- Multiplication of exponents whose powers are the same but bases are different.
|am × bm = (ab)m|
- Division of exponents whose powers are the same, but bases are different.
|am ÷ bm = (a/b) m|
- More than two powers
|(am)n = a m × n|
Using these rules while simplification of expressions with powers can be very easy.
These were the differences between the three terms. To summarise —
- The factors of a number are the numbers which divide the given number completely.
- Multiples of a number are obtained when multiplied by different natural numbers.
- Powers simply represent the number of times the given number is multiplied to itself repeatedly.