3.Playing With Numbers
A number which divides a given number are 1,2,3 and 6; exact divisors or factors of 15 are 1, exactly is called an exact divisor or factor of that 3, 5 and 15. number.
For example exact divisors or factors of 6 are 1, 2, 3 and 6; exact divisors or factors of 15 are 1, 3, 5 and 15.
Factors and Multiples
A factor of a number is an exact divisor of that number. In turn, a number is a multiple of each of its factors. Some interesting facts about factors and multiples are as follows:
- 1 is a factor of every number.
- Every number is a factor of itself.
- Every factor of a number is an exact divisor of that number.
- Every factor of a number is less than or equal to that number.
- The factors of a given number are finite in number.
- Every multiple of a number is greater than or equal to that number.
- The multiples of a given number are infinite in number.
- Every number is a multiple of itself.
Perfect number: If the sum of all the factors of a number is equal to twice the number, then that number is called a perfect number.
For example: 28 is a perfect number because all the factors of 28 are 1,2,4,7,14 and 28 whose sum = 1 + 2 + 4 + 7 + 14 + 28 = 56 = 2 × 28, whereas 10 is not a perfect number because all the factors of 10 are 1, 2, 5 and 10 whose sum = 1 + 2 + 5 + 10 = 18 ≠ 2 × 10.
Prime and Composite Numbers
Prime numbers: The numbers having exactly two factors 1 and the number itself are called prime numbers.
For example, 2, 3, 5, 7, 11, etc. are prime numbers
Composite numbers: The numbers having more than two factors are called composite numbers.
For example 4, 6, 8, 9, 10. etc. are composite numbers.
Note: The number 1 is neither prime nor composite.
Even number: A number which is a multiple of 2 is called an even number.
For example: 2, 4, 6, 8, 10,….
Odd number: A number which is not a multiple of 2 is called an odd number.
For example: 1, 3, 5, 7, 9,…
A number with 0, 2, 4, 6, 8 at the unit’s place is an even number.
So, 250, 2732, 29354, 34596 are even numbers.
Obviously, the numbers 257, 3249, 7321 are odd numbers.
2 is the smallest prime number which is even.
Every prime number except 2 is odd.
Tests for divisibility of numbers Click Here to get the app
Divisibility by 2
If the number ends with 2, 4, 6, 8 or 0, it is divisible by 2.
Example: 28, 54, 96
Here 28, 54 and 96 ends with 8, 4 and 6 respectively.
Therefore, 28, 65 and 96 are divisible by 2.
Divisibility by 3 Click Here to get the app
If the sum of the digits of any number is divisible by 3 then that number is divisible by 3.
Example: 429 ; 4 + 2 + 9 = 15 ; 15 ÷ 3 = 5
Therefore, 429 is divisible by 3.
Divisibility by 4 Click Here to get the app
If the last 2 digits of any number are divisible by 4, then that number is divisible by 4.
Example: 628; Last 2 digits are 28 and 28 ÷ 4 = 7
Therefore, 628 is divisible by 4.
Divisibility by 5 Click Here to get the app
If the digit in the ones place of a number is 5 or 0, then it is divisible by 5.
Example 1: 95 ends in 5;
Therefore, 95 is divisible by 5.
Example 2: 680 ends in 0.
Therefore, 680 is divisible by 5.
Divisibility by 6 Click Here to get the app
If a number is divisible by 2 and 3, then that number is divisible by 6.
Example: 246. It is divisible by 2 as it ends with 6. Now, 2 + 4 + 6 = 12. 12 is divisible by 3, so 246 is divisible by 3 also.
This shows that 246 is divisible by 2 and 3.
Therefore, 246 is divisible by 6.
Divisibility by 8
A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
Example: 2544 Last 3 digits are 544.
544 ÷ 8 = 68
Therefore, 2544 is divisible by 8.
Divisibility by 9
A number is divisible by 9 if the sum of its digits is divisible by 9.
Example: 42,471. 4 + 2 + 4 + 7 + 1 = 18 is divisible by 9.
Therefore 42,471 is divisible by 9.
Divisibility by 10
A number is divisible by 10 if it ends with a ZERO.
Example: 1570. Here the last digit is 0.
Therefore, 1570 is divisible by 10.
Divisibility by 11
A number is divisible by 11 if the difference of the sums of the alternate digits is 0 or a multiple of 11.
Example 1: 9724
9 + 2 = 11. 7 + 4 = 11. Difference between the two sums is 0.
Therefore, 9724 is divisible by 11.
Example 2: 45958
4 + 9 + 8 = 21 ; 5 + 5 = 10 ; 21 – 10 = 11. This is a multiple of 11.
Therefore, 45958 is divisible by 11.
Question: Using divisibility tests, determine whether 1258 is divisible by 6 or not.
Solution: Last digit of the given number is 8. Therefore 1258 is divisible by 2.
When we add all the digits of 1258, the sum is 16. 16 is not divisible by 3
Since the number is not divisible by both 2 and 3, hence it is not divisible by 6.
Excercise 3.2