Important facts and formula:
I.Numeral: In Hindu Arabic System, we use ten symbols 0,1,2,3,4,5,6,7,8,9 called digits to represent any number.
A group of digits representing a number is called a numeral.
We must represent numbers as ones(100 ), tens(101) ,hundreds(102),thousands(103),ten thousands(104 ),lacs(105), ten lakhs or million (106), crores (107 ), ten crores (108).
for example:
in the number 12,34,56,789
9-ones or units place
8-tens place
7-hundreds place
6-thousands place
5-ten thousands place
4-lakhs place
3-ten lakhs or millions place
2-crores place
1-ten crores place
the number is read as twelve crore, thirty-four lakhs ,fifty-six thousand, seven hundred and eighty nine.
II.Place value or Local value of a digit in a number:
Place value of a number is the multiplication of position of the digit where it is in the number and the digit itself, it is like units place or tens place or hundreds place etc..
for example in the number 12,34,56,789
Place value of 9 is 9x1=9;
Place value of 8 is 8x10=80;
Place value of 7 is 7x100=100; ...................
III.Face Value: The face value of a digit itself at whatever place it may be. In the above numeral, the face value of 2 is 2; the face value of 3 is 3 and so on.
IV.Types of Numbers
1.Natural numbers:
All counting numbers 1,2,3,4,5,6,7,8,9....... excluding "0" are called natural numbers.
2.Whole Numbers:
All counting numbers together with zero form the set of whole numbers.
i) 0 is the only whole number which is not a natural number.
ii) Every natural number is a whole number.
3.Integers:
All natural numbers, 0 and negatives of counting numbers i.e., {......-3,-2,-1,0,1,2,3......} together form the set of integers.
i)Positive Intergers: {1,2,3,4,.............} is the set of all positive integers.
ii)Negative Integers: {-1,-2,-3,-4,.....} is the set of negative integers.
iii)Non-positive and Non-negative Integers: 0 is neither positive nor negative. So, {0,1,2,3,.......} represents the set of non-negative integers, while {0,-1,-2,-3,.........} represents the set of non-positive integers.
4.Even Numbers:
A number divisible by 2 is called an even number.
e.g..2,4,6,8,10,......etc..
5.Odd Numbers:
A number not divisible by 2 is called an odd number.
e.g..1,3,5,7,9
6.Prime Numbers:
A number greater than 1 is called a prime number, if it has exactly two factors, namely 1 and the number itself.
prime numbers upto 100 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
7.Composite Numbers:
Numbers greater than 1 which are not prime, are known as composite numbers. e.g.. 4,6,8,9,10,12.
Note:
i) 1 is neither prime nor composite.
ii)2 is the only even number which is prime.
iii)There are 25 prime numbers between 1 and 100.
8.Co-Primes:
Two numbers a and b are said to be co-primes, if their HCF is 1.
e.g..(2,3), (4,5),(7,9) etc.. are co-primes.
TESTS OF DIVISIBILITY 1.Divisibility By 2: A number is divisible by 2, if its unit’ digit is any of 0, 2, 4, 6, 8 Ex.84932 is divisible by 2, while 65935 is not 2. Divisibility By 3: A number is divisible by 3, if the sum of its digits is divisible by 3. Ex. 592482 is divisible by 3, since sum of its digits = (5+9+2+4+8+2) =30, which is divisible by 3. But,864329 is not divisible by 3, since sum of its digits = (8+6+4+3+2+9) =32, which is not divisible by 3. 3. Divisible By 4:A number is divisible by 4, if the number formed by the last two digits is divisible by 4. Ex. 892648 is divisible by 4, since the number formed by the last two digits is 48, which is divisible by4. But, 749282 is not divisible by 4, since the number formed by the last two digits is 82, which is not divisible by 4. 4. Divisibility By 5: A number is divisible by 5, if its unit’s digit is either 0 or 5 Thus, 20820 and 50345 are divisible by 5, while 30934 and 40946 are not. 5. Divisibility By 6: A number is divisible by 6, if it is divisible by both 2 and 3. Ex. The number 35256 is clearly divisible by 2. Sum of its digits = ( 3+5+2+5+6)=21, which is divisible by 3. Thus, 35256 is divisible by 2 as well as 3.Hence, 35256 is divisible by 6. 6.Divisibility by 8: A number is divisible by 8, if the number formed by the last three digits of given number is divisible by 8. Ex:953360 is divisible by 8, since the number formed by last three digits is 360, which is divisible by 8. But, 529418 is not divisible by 8, since the number formed by last three digits is 418, which is not divisible by 8. 7.Divisibility by 9:A number is divisible by 9, if the sum of its digits is divisible by 9. Ex: 60732 is divisible by 9, since sum of digits =(6+0+7+3+2)=18, which is divisible by 9. But, 68956 is not divisible by 9, since sum of digits=(6+8+9+5+6)=34, which is not divisible by 9. 8.Divisibility by 10: A number is divisible by 10, if it ends with 0. Ex:96410, 10480 are divisible by 10, while 96375 is not.
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