Table of Contents
- What is a Multiplication Table?
- Multiplication Table of Numbers from 1 to 10
- Patterns in the Multiplication Table
- Patterns in Multiples of Even and Odd Numbers
- Pattern Related to the Distributive Property of Multiplication
- Pattern Related to the Commutative Property of Multiplication
- Solved Examples
- Frequently Asked Questions
What is a Multiplication Table?
A multiplication table is a table that represents the multiples of two numbers. Familiarizing ourselves with multiplication tables will help us solve math questions easily. The multiples are also known as products and the two numbers that are multiplied are known as factors.
Multiplication Table of Numbers from 1 to 10
The multiplication table of numbers from 1 to 10 can be obtained by multiplying the numbers in the same range by each other. The multiplication table of numbers from 1 to 10 is as follows:
To find the multiple of any number, we need to look at the multiplication table and find the multiple corresponding to that number and the factor with which we are multiplying the number.
What are Patterns in the Multiplication Table?
A pattern is something that can be observed repeatedly in certain situations. Patterns can be observed in the clothes you wear, the flowers in the garden, the ocean, and lots of other things. Similarly, we can observe patterns in the multiplication table as well.
Patterns in Multiples of Even and Odd Numbers
Let’s first consider the product of even numbers.
It is clear that the multiples of even numbers are always even numbers. That is, the products of even numbers always end with 2, 4, 6, 8, or 0.
On the other hand, the multiples of an odd number alternate between odd and even numbers. If an odd number is multiplied by an odd number, the product will also be an odd number. If an odd number is multiplied by an even number, the product will always be an even number.
For example, 5 \( \times \) 5 = 25, 5 \( \times \) 6 = 30, and 5 \( \times \) 7 = 35, where 5 and 7 are odd numbers and 6 is an even number.
There is another pattern hidden among the multiples of odd and even numbers. All products of 4 are double the products of 2. All products of 6 are double the products of 3. Similarly, all products of 8 are twice the products of 4, and all products of 10 are twice the products of 5.
Pattern Related to the Distributive Property of Multiplication
We can observe an interesting pattern in the multiplication table by looking at its columns. Consider the columns for 3, 5, and 8 and compare the products in these columns.
We know that 3 + 5 = 8. An interesting fact here is that the multiples of these numbers follow the same rule. That is, the multiples of 8 are the sum of multiples of 3 and 5 for a given factor. For example, 15, 25, and 40 are multiples of 3, 5, and 8 that we get when we multiply these numbers by 5. And 15 + 25 = 40. Similarly, 27 + 45 = 72, where 27, 45, and 72 are multiples of 3, 5, and 8 that we get by multiplying them with 9.
This pattern is created due to the distributive property of multiplication. The property states that multiplying the sum of two or more addends by a number is the same as multiplying each addend separately by the number and then adding the products together. In math terms, if c = a + b, then 5 \( \times \) c = 5 \( \times \) (a + b) = 5 \( \times \) a + 5 \( \times \) b.
Let’s consider an example. We know that 3 + 5 = 8
Multiplying both sides by 5:
(3 + 5) \( \times \) 5 = 8 \( \times \) 5
3 \( \times \) 5 + 5 \( \times \) 5 = 40
15 + 25 = 40
The same property can be observed for all columns where the sum of values in any two columns is equal to the value of another column for a given row.
Pattern Related to the Commutative Property of Multiplication
Look for a pattern among the highlighted numbers in the diagonals. One of the highlighted diagonals has the numbers 7, 12, 15, 16, 15, 12, and 7. When we reach the middle of the first diagonal, we have 4 \( \times \) 4 = 16. The products following this number are the same as the ones that appeared before it. But they are in reverse order.
The same pattern can be observed in the second and the third highlighted diagonals. The numbers in the second diagonal are 9, 16, 21, 24, 25, 24, 21, 16, and 9, and the numbers in the third diagonal are 20, 27, 32, 35, 36, 35, 32, 27, and 20.
This pattern is created due to the commutative property of multiplication. The commutative property of multiplication states that when we multiply numbers in any order, we will get the same result.
For example, 2 \( \times \) 3 = 6 and 3 \( \times \) 2 = 6, 5 \( \times \) 8 = 40 and 8 \( \times \) 5 = 40. This applies to the multiplication of all numbers.
The numbers in the diagonals repeat themselves in the reverse order because of the factors. Consider the diagonal with the numbers 7, 12, 15, 16, 15, 12, and 7. 16 is obtained by multiplying 4 by itself. On its left side, 15 is obtained by multiplying 5 \( \times \) 3, and on its right side, 15 is obtained by multiplying 3 \( \times \) 5, giving the same result. The same trend is observed among all the numbers in all diagonals.
Solved Examples on Patterns for Multiplication
Example 1: The highlighted row and column of the multiplication table have the same set of numbers. Describe the property that creates this pattern.
Solution:
The property which creates this pattern can be identified by observing the numbers in the highlighted row and column. Consider the number 21 in the highlighted row and the highlighted column. The number 21 in the highlighted column is obtained by multiplying 7 \( \times \) 3, and the 21 in the highlighted row is obtained by multiplying 3 \( \times \) 7.
The commutative property states that 7 \( \times \) 3 = 3 \( \times \) 7 = 21. This pattern is just a variation of the pattern we observed in the diagonals of the table.
Example 2: Compare the rows of 3 and 6. Describe the pattern observed in the multiples of these numbers.
Solution:
We know that 6 = 3 \( \times \) 2.
So, when any number is multiplied by 6, it can be rewritten as that number multiplied by the double of 3. For example, 24 = 6\( \times \) 4 and 24 = 3 \( \times \) 2 \( \times \) 4 = 3 \( \times \) 8.
Therefore, the numbers in the second highlighted row are double the numbers in the first highlighted row.
Example 3: Find the missing number in the following set of numbers. Describe the property observed in this case.
9, 16, __, 24, 25, __, 21, __, 9
Solution:
This set of numbers can be picked up from a diagonal of the multiplication table.
The missing numbers are 21, 24, and 16. So, the complete set of numbers becomes 9, 16, 21, 24, 25, 24, 21, 16, and 9. We can see that the numbers repeat themselves in reverse order after the number 25, which is 5 \( \times \) 5.
This pattern is created due to the commutative property of multiplication.
Example 4: Joel raises a few hens in his backyard. The number of eggs produced by his backyard farm is given below.
Find the number of days required to produce 28 eggs.
Solution:
The number of days are 3, 6, and 9, and the number of eggs produced are 12, 24, and 36. We need to find a number connecting these two sets of numbers.
From the multiplication table, it is clear that we get 12, 24, and 36 when we multiply 4 with 3, 6, and 9, respectively. This means the farm can produce 4 eggs in a day.
To find the number of days required to produce 28 eggs, we just need to find the number that relates to 28 and 4.
Since 7 times 4 is 28, the number of days required to produce 28 eggs is 7.
The product of any number when multiplied by 1 is the number itself. When 1 is one of the factors, the multiples will be 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and so on.
When a number is multiplied by two, we are essentially doubling the number. Also, the product will always be an even number.
The multiples of 5 always end with either 0 or 5. If 5 is multiplied by an even number, the digit in the ones place is always 0. On the other hand, if 5 is multiplied by an odd number, the digit in the ones’ place is always 5.