When a number is divided by its factor, the remainder is zero. A factor of a number completely divides the number.
A number is a multiple of its factors.
E.g. 6×12=12. Therefore 12 is a multiple of both 2 and 6.
Some inferences that can be drawn are:
- 1 is a factor of every number.
- Every number is a factor of itself.
- Every factor is less than or equal to the number.
- A number has a finite number of factors.
- Every number is a multiple of itself.
- Every multiple is greater than or equal to the number.
- Every number has finite number of multiples.
A number whose sum of factors is twice the number is called a perfect number.
E.g. factors of 6= 1,2,3,6. And 1+2+3+6=12. 12 is twice the number 6. Therefore 6 is a perfect number.
PRIME AND COMPOSITE NUMBERS
A prime number is a number which has exactly two factors- 1 and the number itself. E.g. 2, 3, 5, 7, 9, etc.
A composite number is a number which has more than two factors. E.g. 4, 6, 9, 8, etc.
Note: 1 is neither a prime nor a composite number.
TEST FOR DIVISIBILITY OF NUMBERS
- Divisibility by 2: Numbers which have an even number in the ones place are divisible by 2.
- Divisibility by 3: If the sum of the digits of a number is a multiple of 3, then the number is divisible by 3.
- Divisibility by 4: If the numbers formed by the last two digits is divisible by 4, then the number is divisible by 4.
- Divisibility by 5: If the digit at the ones place is 0 or 5, the number is divisible by 5.
- Divisibility by 6: If a number is divisible by both 2 and 3, it is divisible by 6.
- Divisibility by 8: If the number formed by the last three digits is divisible by 8, then the number is divisible by 8.
- Divisibility by 9: If the sum of digits is divisible by 9, the number is said to be divisible by 9.
- Divisibility by 10: If the digit at the ones place is 0, the number is divisible by 10.
- Divisibility by 11: if the difference between the sum of digits at odd places(from the right) and the sum of digits at even places(from the left) is divisible by 11, the number is said to be divisible by 11.
Consider two numbers 6 and 15.
Factors of 6= 1, 2, 3, 6
Factors of 15= 1, 3, 5, 15
1 and 3 are the common factors of 6 and 15.
If two numbers have only 1 as a common factor, they are called co-prime numbers.
E.g. Consider 5 and 16.
Factors of 5= 1, 5
Factors of 16= 1, 2, 4, 8, 16
Common factor= 1
Therefore, 5 and 16 are co-prime numbers.
If a number is a multiple of two or more numbers, it is said to be their common multiple.
SOME DIVISIBILITY RULES
- If a number is divisible by another number then it is also divisible by all the factors of that number.
- A number that is divisible by two co-prime numbers is also divisible by their product.
- If two given numbers are divisible by a number then their sum and difference is also divisible by that number.
When a number is expressed as a product of its prime factors, it is called prime factorisation.
E.g. 12= 2x2x3 and 28= 2x2x7
HIGHEST COMMON FACTOR (HCF)
The HCF of two or more numbers is the highest of their common factors of the number. It is also called Greatest Common Divisor (GCD).
HCF can be calculated using prime factorization.
E.g. HCF of 20, 24, 36
We take the common factor i.e. 2 and it occurs twice therefore, HCF= 2×2= 4
LOWEST COMMON MULTIPLE (LCM)
The lowest among the common multiple of two or more numbers is called their LCM.
Finding the LCM of 16 and 20 using prime factorization:
LCM is the product of the prime factors counted the number of times they occur.
Here, LCM= 2x2x2x2x5= 80