Introduction:
Natural Numbers:
All positive integers are natural numbers.
Ex 1,2,3,4,8,......
There are infinite natural numbers and number 1 is the least natural number.
Based on divisibility there would be two types of natural numbers. They are Prime and composite.
Prime Numbers:
A natural number larger than unity is a prime number if it does not have other divisors except for itself and unity.
Note:-Unity i e,1 is not a prime number.
Properties Of Prime Numbers:
->The lowest prime number is 2.
->2 is also the only even prime number.
->The lowest odd prime number is 3.
->The remainder when a prime number p>=5 s divided by 6 is 1 or 5.However, if a number on being divided by 6 gives a remainder 1 or 5 need not be prime.
->The remainder of division of the square of a prime number p>=5 divide by 24 is 1.
->For prime numbers p>3, p²-1 is divided by 24.
->If a and b are any 2 odd primes then a²-b² is composite. Also a²+b² is composite.
->The remainder of the division of the square of a prime number p>=5 divided by 12 is 1.
Process to Check A Number s Prime or not:
Take the square root of the number.
Round of the square root to the next highest integer call this number as Z.
Check for divisibility of the number N by all prime numbers below Z. If there is no numbers below the value of Z which divides N then the number will be prime.
Example 239 is prime or not?
√239 lies between 15 or 16.Hence take the value of Z=16.
Prime numbers less than 16 are 2,3,5,7,11 and 13.
239 is not divisible by any of these. Hence we can conclude that 239
is a prime number.
Composite Numbers:
The numbers which are not prime are known as composite numbers.
Co-Primes:
Two numbers a an b are said to be co-primes,if their H.C.F is 1.
Example (2,3),(4,5),(7,9),(8,11).....
Place value or Local value of a digit in a Number:
place value:
Example 689745132
Place value of 2 is (2*1)=2
Place value of 3 is (3*10)=30 and so on.
Face value:-It is the value of the digit itself at whatever
place it may be.
Example 689745132
Face value of 2 is 2.
Face value of 3 is 3 and so on.
Tests of Divisibility:
Divisibility by 2:-A number is divisible by 2,if its unit's digit is any of 0,2,4,6,8.
Example 84932 is divisible by 2,while 65935 is not.
Divisibility by 3:-A number is divisible by 3,if the sum of its digits is divisible by 3.
Example 1.592482 is divisible by 3,since sum of its digits
5+9+2+4+8+2=30 which is divisible by 3.
Example 2.864329 is not divisible by 3,since sum of its digits
8+6+4+3+2+9=32 which is not divisible by 3.
Divisibility by 4:-A number is divisible by 4,if the number formed by last
two digits is divisible by 4.
Example 1.892648 is divisible by 4,since the number formed by the last
two digits is 48 divisible by 4.
Example 2.But 749282 is not divisible by 4,since the number formed by
the last two digits is 82 is not divisible by 4.
Divisibility by 5:-A number divisible by 5,if its unit's digit is either
0 or 5.
Example 20820,50345
Divisibility by 6:-If the number is divisible by both 2 and 3.
example 35256 is clearly divisible by 2
sum of digits =3+5+2+5+21,which is divisible by 3
Thus the given number is divisible by 6.
Divisibility by 8:-A number is divisible by 8 if the last 3 digits
of the number are divisible by 8.
Divisibility by 11:-If the difference of the sum of the digits in the
odd places and the sum of the digitsin the even places is zero or divisible
by 11.
Example 4832718
(8+7+3+4) - (1+2+8)=11 which is divisible by 11.
Divisibility by 12:-All numbers divisible by 3 and 4 are divisible by 12.
Divisibility by 7,11,13:-The difference of the number of its thousands
and the remainder of its division by 1000 is divisible by 7,11,13.
BASIC FORMULAE:
->(a+b)²=a²+b²+2ab
->(a-b)²=a²+b²-2ab
->(a+b)²-(a-b)²=4ab
->(a+b)²+(a-b)²=2(a²+b²)
->a²-b²=(a+b)(a-b)
->(a-+b+c)²=a²+b²+c²+2(ab+b c+ca)
->a³+b³=(a+b)(a²+b²-ab)
->a³-b³=(a-b)(a²+b²+ab)
->a³+b³+c³-3a b c=(a+b+c)(a²+b²+c²-ab-b c-ca)
->If a+b+c=0 then a³+b³+c³=3a b c
DIVISION ALGORITHM
If we divide a number by another number ,then
Dividend = (Divisor * quotient) + Remainder
MULTIPLICATION BY SHORT CUT METHODS
1.Multiplication by distributive law:
a)a*(b+c)=a*b+a*c
b)a*(b-c)=a*b-a*c
Example
a)567958*99999=567958*(100000-1)
567958*100000-567958*1
56795800000-567958
56795232042
b)978*184+978*816=978*(184+816)
978*1000=978000
2.Multiplication of a number by 5n:-
Put n zeros to the right of the multiplicand and divide the number so formed by 2n
Example 975436*625=975436*54=9754360000/16=609647500.
PROGRESSION:
A succession of numbers formed and arranged in a definite order according to certain definite rule is called a progression.
1.Arithmetic Progression:-If each term of a progression differs from its preceding term by a constant.
This constant difference is called the common difference of the A.P.
The n th term of this A.P is Tn=a(n-1)+d.
The sum of n terms of A.P Sn=n/2[2a+(n-1)d].
xImportant Results:
a.1+2+3+4+5......................=n(n+1)/2.
b.12+22+32+42+52......................=n(n+1)(2n+1)/6.
c.13+23+33+43+53......................=n2(n+1)2/4
2.Geometric Progression:-A progression of numbers in which every
Natural Numbers:
All positive integers are natural numbers.
Ex 1,2,3,4,8,......
There are infinite natural numbers and number 1 is the least natural number.
Based on divisibility there would be two types of natural numbers. They are Prime and composite.
Prime Numbers:
A natural number larger than unity is a prime number if it does not have other divisors except for itself and unity.
Note:-Unity i e,1 is not a prime number.
Properties Of Prime Numbers:
->The lowest prime number is 2.
->2 is also the only even prime number.
->The lowest odd prime number is 3.
->The remainder when a prime number p>=5 s divided by 6 is 1 or 5.However, if a number on being divided by 6 gives a remainder 1 or 5 need not be prime.
->The remainder of division of the square of a prime number p>=5 divide by 24 is 1.
->For prime numbers p>3, p²-1 is divided by 24.
->If a and b are any 2 odd primes then a²-b² is composite. Also a²+b² is composite.
->The remainder of the division of the square of a prime number p>=5 divided by 12 is 1.
Process to Check A Number s Prime or not:
Take the square root of the number.
Round of the square root to the next highest integer call this number as Z.
Check for divisibility of the number N by all prime numbers below Z. If there is no numbers below the value of Z which divides N then the number will be prime.
Example 239 is prime or not?
√239 lies between 15 or 16.Hence take the value of Z=16.
Prime numbers less than 16 are 2,3,5,7,11 and 13.
239 is not divisible by any of these. Hence we can conclude that 239
is a prime number.
Composite Numbers:
The numbers which are not prime are known as composite numbers.
Co-Primes:
Two numbers a an b are said to be co-primes,if their H.C.F is 1.
Example (2,3),(4,5),(7,9),(8,11).....
Place value or Local value of a digit in a Number:
place value:
Example 689745132
Place value of 2 is (2*1)=2
Place value of 3 is (3*10)=30 and so on.
Face value:-It is the value of the digit itself at whatever
place it may be.
Example 689745132
Face value of 2 is 2.
Face value of 3 is 3 and so on.
Tests of Divisibility:
Divisibility by 2:-A number is divisible by 2,if its unit's digit is any of 0,2,4,6,8.
Example 84932 is divisible by 2,while 65935 is not.
Divisibility by 3:-A number is divisible by 3,if the sum of its digits is divisible by 3.
Example 1.592482 is divisible by 3,since sum of its digits
5+9+2+4+8+2=30 which is divisible by 3.
Example 2.864329 is not divisible by 3,since sum of its digits
8+6+4+3+2+9=32 which is not divisible by 3.
Divisibility by 4:-A number is divisible by 4,if the number formed by last
two digits is divisible by 4.
Example 1.892648 is divisible by 4,since the number formed by the last
two digits is 48 divisible by 4.
Example 2.But 749282 is not divisible by 4,since the number formed by
the last two digits is 82 is not divisible by 4.
Divisibility by 5:-A number divisible by 5,if its unit's digit is either
0 or 5.
Example 20820,50345
Divisibility by 6:-If the number is divisible by both 2 and 3.
example 35256 is clearly divisible by 2
sum of digits =3+5+2+5+21,which is divisible by 3
Thus the given number is divisible by 6.
Divisibility by 8:-A number is divisible by 8 if the last 3 digits
of the number are divisible by 8.
Divisibility by 11:-If the difference of the sum of the digits in the
odd places and the sum of the digitsin the even places is zero or divisible
by 11.
Example 4832718
(8+7+3+4) - (1+2+8)=11 which is divisible by 11.
Divisibility by 12:-All numbers divisible by 3 and 4 are divisible by 12.
Divisibility by 7,11,13:-The difference of the number of its thousands
and the remainder of its division by 1000 is divisible by 7,11,13.
BASIC FORMULAE:
->(a+b)²=a²+b²+2ab
->(a-b)²=a²+b²-2ab
->(a+b)²-(a-b)²=4ab
->(a+b)²+(a-b)²=2(a²+b²)
->a²-b²=(a+b)(a-b)
->(a-+b+c)²=a²+b²+c²+2(ab+b c+ca)
->a³+b³=(a+b)(a²+b²-ab)
->a³-b³=(a-b)(a²+b²+ab)
->a³+b³+c³-3a b c=(a+b+c)(a²+b²+c²-ab-b c-ca)
->If a+b+c=0 then a³+b³+c³=3a b c
DIVISION ALGORITHM
If we divide a number by another number ,then
Dividend = (Divisor * quotient) + Remainder
MULTIPLICATION BY SHORT CUT METHODS
1.Multiplication by distributive law:
a)a*(b+c)=a*b+a*c
b)a*(b-c)=a*b-a*c
Example
a)567958*99999=567958*(100000-1)
567958*100000-567958*1
56795800000-567958
56795232042
b)978*184+978*816=978*(184+816)
978*1000=978000
2.Multiplication of a number by 5n:-
Put n zeros to the right of the multiplicand and divide the number so formed by 2n
Example 975436*625=975436*54=9754360000/16=609647500.
PROGRESSION:
A succession of numbers formed and arranged in a definite order according to certain definite rule is called a progression.
1.Arithmetic Progression:-If each term of a progression differs from its preceding term by a constant.
This constant difference is called the common difference of the A.P.
The n th term of this A.P is Tn=a(n-1)+d.
The sum of n terms of A.P Sn=n/2[2a+(n-1)d].
xImportant Results:
a.1+2+3+4+5......................=n(n+1)/2.
b.12+22+32+42+52......................=n(n+1)(2n+1)/6.
c.13+23+33+43+53......................=n2(n+1)2/4
2.Geometric Progression:-A progression of numbers in which every
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