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<body bgcolor="aqua"><center><h2>Numbers</h2></center>
<pre>
<font size="2">
<strong>Introduction:</strong>
<strong>Natural Numbers:</strong>
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
<strong>Prime and composite.</strong>
<b>Prime Numbers:</b>
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.
<b>Properties Of Prime Numbers:</b>
->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.
<b>Process to Check A Number s Prime or not:</b>
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.
<u>Example 239 is prime or not?</u>
√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.
<b>Composite Numbers:</b>
The numbers which are not prime are known as composite numbers.
<strong>Co-Primes:</strong>
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:
<strong>place value:</strong>
<u>Example 689745132</u>
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.
<u>Example 689745132</u>
Face value of 2 is 2.
Face value of 3 is 3 and so on.
<font size="3"> <a href="numbers.html"><b>Top</b></a></font>
<strong>Tests of Divisibility:</strong>
Divisibility by 2:-A number is divisible by 2,if its unit's digit is
any of 0,2,4,6,8.
<u>Example 84932 is divisible by 2,while 65935 is not.</u>
Divisibility by 3:-A number is divisible by 3,if the sum of its digits is
divisible by 3.
<u>Example 1.</u>592482 is divisible by 3,since sum of its digits
5+9+2+4+8+2=30 which is divisible by 3.
<u>Example 2.</u>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.
<u>Example 1.</u>892648 is divisible by 4,since the number formed by the last
two digits is 48 divisible by 4.
<u>Example 2.</u>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.
<u>Example</u> 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.
<u>Example</u> 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.
<strong>BASIC FORMULAE:</strong>
->(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
<strong>DIVISION ALGORITHM</strong>
If we divide a number by another number ,then
Dividend = (Divisor * quotient) + Remainder
<font size="3"> <a href="numbers.html"><b>Top</b></a></font>
<strong>MULTIPLICATION BY SHORT CUT METHODS</strong>
1.Multiplication by distributive law:
a)a*(b+c)=a*b+a*c
b)a*(b-c)=a*b-a*c
<u>Example</u>
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
<u>Example</u> 975436*625=975436*54=9754360000/16=609647500.
<strong>PROGRESSION:</strong>
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
term bears a constant ratio with ts preceding term.
i.e a,a r,a r2,a r3...............
In G.P Tn=a r n-1
Sum of n terms Sn=a(1-r n)/1-r
<strong>Problems</strong>
1.Simplify
a.8888+888+88+8
b.11992-7823-456
Solution: a.8888
888
88
8
9872
b.11992-7823-456=11992-(7823+456)
=11992-8279=3713
2.What could be the maximum value of Q in the following equation?
5PQ+3R7+2Q8=1114
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