Basic concepts, definitions and identities

Number System


Test of divisibility:
1. A number is divisible by ‘2’ if it ends in zero or in a digit which is a multiple of
‘2’i.e. 2,4, 6, 8.

2. A number is divisible by ‘3’, if the sum of the digits is divisible by ‘3’.

3. A number is divisible by ‘4’ if the number formed by the last two digits, i.e. tens
and units are divisible by 4.

4. A number is divisible by ‘5’ if it ends in zero or 5

5. A number is divisible by ‘6’ if it divisible by ‘2’ as well as by ‘3’.

6. A number is divisible by ‘8’ if the number formed by the last three digits, i.e,
hundreds tens and units is divisible by ‘8’.

7. A number is divisible by ‘9’ if the sum of its digit is divisible by ‘9’

8. A number is divisible by ‘10’ if it ends in zero.

9. A number is divisible by ‘11’ if the difference between the sums of the digits in
the even and odd places is zero or a multiple of ‘11’.

LCM:
LCM of a given set of numbers is the least number which is exactly divisible
by every number of the given set.

HCF:
HCF of a given set of numbers is the highest number which divides exactly
every number of the given set
.
LCM, HCF:

1. Product of two numbers = HCF × LCM

2. HCF of fractions =LCM of denominators/HCF of numerators

3. LCM of fractions =HCF of denominators/LCM of numerators

4. One number =(LCM × HCF)/2nd Number
 
5. LCM of two numbers=Product of the numbers/HCF

6.  HCF =Product of the numbers/LCM


NOTES:
 1. The square of an odd number is always odd.
2. A number is said to be a prime number if it is divisible only by itself and unity.
Ex. 1, 2, 3, 5,7,11,13 etc.
3. The sum of two odd number is always even.
4. The difference of two odd numbers is always even.
5. The sum or difference of two even numbers is always even.
6. The product of two odd numbers is always odd.
7. The product of two even numbers is always even.

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