Chapter 3 - Playing with numbers

Playing with Numbers (Introduction)

Multiplication as Repeated Addition

Multiplication is repeated addition. If a number is getting repeatedly added several times, then you can multiply it simply by the number of times it is being added.

Example

Adding 5 to itself 8 times to get 40 is same as multiplying 5 by 8 to get 40.
      5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 40
Or,
      5 × 8 = 40

Factors and Multiples

Let us re-visit the multiplication table:

5 × 1 = 5

5 × 2 = 10

5 × 3 = 15

The numbers on the left (of equal sign) are called the factors and the ones on the right are called the multiples.

For example, in case of

5 × 3 = 15

5 and 3 are factors and 15 is the multiple.

Division as Repeated Subtraction

Division is repeated subtraction.

Example

If 4 is subtracted from 16 repeatedly for 4 times, we get zero at the end.
This is same as dividing 16 by 4.
      16 − 4 − 4 − 4 − 4 = 0
Or,
      16 ÷ 4 = 0

Exact division of a number (dividend) is the continuous subtraction of a number (divisor) till zero (remainder) is obtained.

Factors and Multiples

What are Multiples?

The multiples of X are the resulting numbers obtained by multiplying X by other numbers (1, 2, 3, 4, 5...). The multiples of any number are infinitely many.

Example

Consider the multiplication table of 3:
         3 × 1 = 3
         3 × 2 = 6
         3 × 3 = 9
         3 × 4 = 12
         3 × 5 = 15
         ...........

The resulting numbers obtained by multiplying 3 with different numbers, i.e., 3, 6, 9, 12, and 15 are called multiples of 3.

What are Factors?

The factor of X is a number that perfectly divides X. For example, 3 is a factor of 12 as it divides 12 perfectly (leaving no remainder)

Example

Which of the two, 5 or 7, is a factor of 30?
                  30 ÷ 5
                  30 ÷ 7
In the first case,
                  30 ÷ 5 = 6
5 perfectly divides 30 with 0 as remainder and 6 as quotient. So 5 is a factor of 30.
In the second case,
                  30 ÷ 7
Produces a remainder 2. As 7 does not perfectly divide 30, it is not a factor of 30.

Multiples are Opposite of Factors

Let's take an example:

3 × 9 = 27

In this case, 27 is a multiple of both 3 and 9.

At the same time, 3 and 9 are factors of 27, as they both perfectly divide 27.

Interesting Facts

Every number is a multiple of itself.
Every number is a factor of itself
Factors of a number are finite, while there can be infinite multiples of a number.
Every factor of a number is less than or equal to that number.
Every multiple of a number is greater than or equal to that number.
1 is a common factor of all numbers.

Prime Numbers and Composite Numbers

Prime Numbers

The numbers that have only two factors are called prime numbers. In other words, a prime number can be divisible by 1 and itself.

For example, 2, 3, 5, 7, 11... are prime numbers. These numbers have only two factors: 1 and themselves.

Composite Numbers

The numbers that have more than two factors are called composite numbers. For example, 4, 6, 8, 9, 10... are composite numbers.

Note: The number 1 is neither prime nor composite.

Sieve of Eratosthenes

For finding prime and composite numbers between 1 and 100, the concept of multiples rather than factors is going to be used.

A Greek mathematician Eratosthenes used the concept of multiples to filter or sieve out the prime and composite numbers between 1 to 100.

In the list of numbers, 1 is neither prime nor composite, so 1 is left out.
We go to the number next to 1, i.e., 2. All the multiples of 2 in the list like 4, 6, 8, 10, 12... are crossed out.
We go to the number next to 2, i.e., 3 which is not crossed out. All the multiples of 3 like 6, 9, 12, 15... are crossed out.
We move to the next number that is not crossed out, i.e., 5. All the multiples of 5 like 10, 15, 20, 25... are crossed out.
Note that after 3, we jump to 5 because 4 is already crossed out from the list, as it is a multiple of 2.
This process is repeated till all the numbers are either encircled or crossed out.

The numbers that are circled are prime numbers and all the numbers that are crossed out are composite numbers.

Even Numbers and Odd Numbers

Even Numbers

All those numbers that are divisible by 2 or are multiples of 2 are called even numbers.

Let us now learn about another classification of numbers based on the end-digit of the number or the digit in one's place.

Consider some multiples of 2:

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22 ....

We see a pattern here. All these numbers have 0, 2, 4, 6, 8 as end-digits.

So, all those numbers that end with either with 2, 4, 6, 8, or 0 are called as even numbers.

Example

450 is even, as it ends in a 0.
4986 is even number, as it ends in a 6.
What about 9,99,998? This too is even, as it has 8 at the end.

Odd Numbers

Numbers that end with 1, 3, 5, 7, or 9 are called odd numbers.

Odd numbers are also defined as those numbers that are not divisible by 2 or that are not multiples of 2.

Consider the number series:

1, 3, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25 ...

Here we see the pattern that the numbers end with either 1, 3, 5, 7 or 9. Such numbers, as we have already defined, are called odd numbers.

Example

789 is odd, as it ends with 9.
Similarly, 4515 is odd, as it ends with 5.
Let's take 673421821. This too is odd, as it ends with 1.

Tests for Divisibility of Numbers

Divisibility Rule for 2

A number is said to be divisible by 2, if it ends with a 0, 2, 4, 6, or 8.

Example

Consider the number, 486.
Since its last digit is 6, it is divisible by 2.
Consider another number, 789.
Its last digit is 9. So, 789 is not divisible by 2.

Divisibility Rule for 3 and 9

The divisibility rules for 3 and 9 are similar.

Sum up the digits of the number.
If the sum is a multiple of 3, then the number itself is divisible by 3.

Example

Consider the number, 486.
Sum of digits, 4 + 8 + 6 = 18
18 is a multiple of 3.
So, 486 is divisible by 3.
18 is a multiple of 9. So, 486 is divisible by 9 too.

Now, consider another number, 29.
Sum of digits, 2 + 9 = 11
11 is neither a multiple of 3 nor 9.
So, 29 is neither divisible by 3 nor 9.

Divisibility Rule for 6

If a number is divisible by both 2 and 3, it is divisible by 6.

Example

486 is divisible by both 2 and 3.
Hence, 486 is divisible by 6 too!

Let's take 488.
It's divisible by 2, but not by 3.
So, 488 is not divisible by 6.

Divisibility Rule for 4

A number with 2 or more digits is divisible by 4 if the number formed by the last two digits (last from left) is divisible by 4.

Example

Consider 552.
52 is divisible by 4.
So, 552 is divisible by 4.

What about 8261?
61 is not divisible by 4.
So, 8261 is not divisible by 4.

Divisibility Rule for 8

A number with 3 or more digits is divisible by 8 if the number formed by the last three digits (last from left) is divisible by 8.

Example

Consider 5,55,552.
The number formed by last three digits is 552.
552 is divisible by 8.
So, 5,55,552 is divisible by 8.

Divisibility Rule for 5

If a number ends either in a 0 or a 5, then it is divisible by 5. For example, 790 and 795 are divisible by 5.

Divisibility Rule for 10

This is the easiest divisibility rule. If a number ends in a 0 only, then it is divisible by 10. For example, 790 is divisible by 10, but 795 is not.

Divisibility Rule for 11

If the difference of the sums of digits in even and odd places of a number is either a 0 or a multiple of 11, the number is divisible by 11.

Example

Consider the number, 4565.
Sum of digits in even places = 6 + 4 = 10
Sum of digits in odd places = 5 + 5 = 10
Difference of sums = 10 − 10 = 0
So, the number 4565 is divisible by 11.

Common Factors and Common Multiples

Common Factors

Factors are the numbers that perfectly divide a given number. For example, the factors of 8 are 1, 2, 4, and 8.

Two or more numbers can have many things in common; for example, factors.

Example

Consider the numbers, 15 and 20.
Factors of 15 are: 1, 3, 5, 15
Factors of 20 are 1, 2, 4, 5, 10, 20
The common factors of 15 and 20 are 1 and 5. Factor 1 is trivial, as it a factor of every number.

Example

Consider the numbers, 12 and 24
Factors of 12 are: 1, 2, 3, 4, 6, 12
Factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24
The common factors of these two numbers are 1, 2, 3, 4, 6, and 12.

Co-prime Numbers

Numbers that have only 1 as their common factor are called co-prime numbers. For example, 9 and 10 are co-prime numbers.

Similarly, 215 and 216 are co-prime numbers, as the only common factor they have is 1.

Common Multiples

Multiples of a number (X) are the numbers that we get when we multiply X with other numbers (like 1, 2, 3, 4...). For example, 8, 16, 24, 32... are multiples of 8.

Example

Consider the multiples of the numbers, 4 and 8.
Multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32...
Multiples of 8 are: 8, 16, 24, 32...
The common multiples of 4 and 8 are 8, 16, 24....
Common factors and common multiples are useful in understanding the concept of LCM and HCF.

Prime Factorisation

What is Prime Factorization?

It is also known that composite numbers can be expressed as products of their factors.

For example, the composite number 20 can be expressed as,

4 × 5 = 20

2 × 10 = 20

The factor 4 can further be expressed as

2 × 2 = 4

Similarly, 10 can be further factorized as

2 × 5 = 10

In this way, we express the composite number as product of its prime factors. For example,

36 = 2 × 2 × 3 × 3

And,

45 = 3 × 3 × 5

This process of breaking down a number into its prime factors is called prime factorization.

Example

Question: Find the prime factorization of 4030.
Solution: Using divisibility rules, we know that
   4030 is divisible by 2
   4030 ÷ 2 = 2015
2015 is not divisible by 3 because
   2 + 0 + 1 + 5 = 8
And, 8 is not divisible by 3.
However, 2015 is divisible by 5, as it ends with 5.
   2015 ÷ 5 = 403
Trying out different numbers, it is found that 403 is divisible by 13.
   403 ÷ 13 = 31
13 and 31 are prime numbers. Factorization ends here.
Hence, we can express 4030 as,
   4030 = 2 × 5 × 13 × 31
2, 5, 13, and 31 are the prime factors of 4030.

Factor Trees

A number can be expressed in terms of its prime factors using the concept of factor trees.

The factor tree for 4030 would look as follows:

HCF by Prime Factorisation

Definition: HCF or highest common factor of two or more numbers is the highest number that perfectly divides those numbers.

We can use the prime factorization method to find the HCF of two or more numbers.

Example

Question: Find the HCF of 5, 10, and 15.
Solution:
   5 = 1 × 5
   10 = 2 × 5
   15 = 3 × 5
The common factor among these is 5 and it is the highest as well. Hence 5 is the HCF of 5, 10, and 15.

Example

Question: Find the HCF of 4, 8, and 16.
Solution:
   4 = 2 × 2
   8 = 2 × 2 × 2
   16 = 2 × 2 × 2 × 2
The HCF of these three numbers is = 2 × 2 = 4.

Example

Find the HCF of the numbers 14, 15, and 16.
Solution:
   14 = 2 × 7
   15 = 3 × 5
   16 = 2 × 2 × 2 × 2
There is no common factor except 1 for these numbers. Hence, 1 is the HCF of 14, 15, and 16.

HCF by Division Method

Besides prime factorization, there is another method of finding the HCF of two or more number. It is known as the division method.

Let's understand this method with examples.

Example: Find the HCF of 8, 28, and 32.

Solution: Let's take the first two numbers, 8 and 28.

Divide 28 by 8 → remainder is 4.

Next, bring the divisor (8) down and divide it with the remainder (4).

Repeat the process till you get 0 as the remainder.

Note down the divisor that produces 0 as the remainder.

The HCF of 8 and 28 is 4.

Next, take the third number 32 and divide it with 4.

We get 0 as the remainder.

So, 4 is the HCF of 8, 28, and 32.

Understanding LCM

Definition: LCM is the least common multiple among all the multiples of two or more numbers.

There are two methods to find the LCM of a set of numbers:

The Ladder Method
The Prime Factorization Method

The Ladder Method

Write the set of numbers, whose LCM is to be found out, in a comma-separated sequence.
Divide the set of numbers with the smallest possible prime number that divides the maximum of numbers.
The numbers that are not divisible are brought down as remainder.
Again, select a prime number as the divisor and divide all the remainders.
Repeat the process till the numbers do not have any common factors.
A ladder of numbers will be formed during the division process.
Then the product of the numbers outside the ladder gives the LCM of the given numbers.

Example: Find the LCM of 4, 9, and 16 using the ladder method.

The least prime number 2 is taken and it is seen that it divides two of the three given numbers i.e., 4 and 16. Since 9 is indivisible by 2 it is brought down as it is. Again a prime number (2) that divides the numbers is taken as divisor and division is done as follows. The numbers 1, 9 and 4 are obtained. At least two numbers with the same factor are not there.

Then the product of all the numbers outside the ladder gives the LCM

So the LCM = 2 × 2 × 9 × 4 = 144

Prime Factorization Method

Every number can be expressed as a product of one or more prime numbers.

In this method, the LCM of a set of numbers is found by using the prime factors of each of those numbers.

The maximum number of times each of the prime factor occurs is found and the product of such numbers gives the LCM.

Example

Question: Find the LCM of 6, 9, and 15.
Solution: 
In prime factorization method, the numbers are written as product of their prime factors.
   6 = 2 × 3
   9 = 3 × 3
   15 = 3 × 5
Prime factor 2 occurs only once in 6.
The maximum number of times 2 occurs is only 1.
Prime factor 3 occurs in all the three numbers, but it occurs twice in 9. It's the maximum. So, 3 is taken twice.
Prime factor 5 occurs only once in 15.
So, the LCM of 6, 9, and 15 is,
   2 × 3 × 3 × 5 = 90

Example

Question: Find the LCM of 4, 8, and 16.
Solution: Product of prime factors,
   4 = 2 × 2
   8 = 2 × 2 × 2
   16 = 2 × 2 × 2 × 2
In all the given numbers, there is only one prime factor, i.e., 2.
In 16, the prime factor 2 occurs for the maximum number of times, i.e., 4.
So, the LCM of 4, 8, and 16 is,
2 × 2 × 2 × 2 = 16