1. (a+b)2=a2+b2+2ab
2. (a−b)2=a2+b2−2ab
3. (a+b)2−(a−b)2=4ab
4. (a+b)2+(a−b)2=2(a2+b2)
5. (a2–b2)=(a+b)(a−b)
6. (a+b+c)2=a2+b2+c2+2(ab+bc+ca)
7. (a3+b3)=(a+b)(a2−ab+b2)
8. (a3–b3)=(a−b)(a2+ab+b2)
9. (a3+b3+c3−3abc)=(a+b+c)(a2+b2+c2−ab−bc−ca)
10. If a+b+c=0, then a3+b3+c3=3abc.
Types
of Numbers:
I. Natural Numbers:
Counting
numbers 1,2,3,4,5,…… are called natural numbers
II. Whole Numbers:
All
counting numbers together with zero form the set of whole numbers.
Thus,
(i) 0 is the only whole number which is not a natural number.
(ii) Every natural number is a whole number.
III. 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 Integers: 1,2,3,4,….. is the set of all positive
integers.
(ii) Negative Integers: −1,−2,−3,….. is the set of all 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.
IV. Even Numbers:
A
number divisible by 2 is called an even number, e.g.,2,4,6,8, etc.
V. Odd Numbers:
A
number not divisible by 2 is called an odd number. e.g.,1,3,5,7,9,11, etc.
VI. 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 up to 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.
·
Prime
numbers Greater than 100 : Let p be a given number greater than 100. To find
out whether it is prime or not, we use the following method :
Find
a whole number nearly greater than the square root of p. Let k>*jp.
Test whether p is divisible by any prime number less than k. If yes, then p is
not prime. Otherwise, p is prime. Example: We have to find
whether 191 is a prime number or not. Now, 14>V191.
Prime numbers less than 14 are 2,3,5,7,11,13.
191 is not divisible by any of them. So, 191 is a prime number.
VII. 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.
Remainder
& Quotient:
"The remainder is r when p is divided by k"
means p=kq+r the integer q is called the quotient.
For instance, "The remainder is 1 when 7 is divided by 3"
means 7=3*2+1. Dividing both sides of p=kq+r by k gives the
following alternative form
Example: The remainder is 57 when a number is divided by
10,000. What is the remainder when the same number is divided by 1,000?
(A)
5
(B)
7
(C)
43 (D)
57
(E)
570
Solution:
Since
the remainder is 57 when the number
is divided by 10,000, the number can
be expressed as 10,000n+57, where n is an integer.
Rewriting 10,000 as 1,000*10 yields 10,000n+57=1,000(10n)+57
Now, since n is an integer, 10 n is an integer.
Letting 10n=q , we get
10,000n+57=1,000*q+57
Hence, the remainder is still 57 (by the p=kq+r form) when the
number is divided by 1,000. The answer is (D).
Even,
Odd Numbers:
A number n is even if the remainder is zero when n is divided by 2:n=2z+0,
orn=2z.
A number n is odd if the remainder is one when n is divided by 2:n=2z+1.
The following properties for odd and even numbers are very useful—you shouldmemorize
them:
even
* even = even
odd * odd = odd
even * odd = even
even + even = even
odd + odd = even
even + odd = odd
Example: If n is a positive integer and (n + 1)(n + 3) is odd,
then (n + 2)(n + 4) must be a multiple of which one of the following?
(A) 3
(B) 5
(C)
6 (D)
8
(E) 16
Solution:
(n+1)(n+3) is
odd only when both (n+1) and (n+3) are odd. This is
possible only when n is even.
Hence, n = 2m, where m is a positive integer. Then,
(n+2)(n+4)=(2m+2)(2m+4)=2(m+1)2(m+2)=4(m+1)(m+2)=
4 * (product of two consecutive positive integers, one which must be even)
= 4 * (an even number), and this equals a number that is at least a
multiple of 8
Hence, the answer is (D).
Tests
of Divisibility:
1. Divisibility By 2:
A
number is divisible by 2, if its unit's 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. Divisibility 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 by 4.
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 the
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.
9.
Divisibility By 11:
A
number is divisible by 11, if the difference of the sum of its digits at odd
places and the sum of its digits at even places, is either 0 or a number
divisible by 11.
Ex. The number 4832718 is divisible by 11, since :(sum of
digits at odd places) - (sum of digits at even places) =
=(8+7+3+4)−(1+2+8)=11, which is divisible by 11.
10. Divisibility By 12:
A
number is divisible by 12, if it is divisible by both 4 and3.
Ex. Consider the number 34632.
(i)
The number formed by last two digits is 32, which is divisible by 4,
(ii) Sum of digits =(3+4+6+3+2)=18, which is divisible by 3. Thus, 34632
is divisible by 4 as well as 3. Hence, 34632 is divisible by 12.
11. Divisibility By 14:
A
number is divisible by 14, if it is divisible by 2 as well as 7.
12. Divisibility By 15:
A
number is divisible by 15, if it is divisible by both 3 and 5.
13. Divisibility By 16:
A
number is divisible by 16, if the number formed by the last4 digits is
divisible by 16.
Ex.7957536 is divisible by 16, since the number formed by the last four
digits is 7536, which is divisible by 16.
14. Divisibility By
24:
A
given number is divisible by 24, if it is divisible by both 3 and 8.
15. Divisibility By
40:
A
given number is divisible by 40, if it is divisible by both 5 and 8.
16. Divisibility By
80:
A
given number is divisible by 80, if it is divisible by both 5 and 16.
Note: If a number is divisible by p as well as q, where p and q are
co-primes, then the given number is divisible by pq. If p and
q are not co-primes, then the given
number need not be divisible by pq, even when it is divisible
by both p and q.
Ex. 36 is divisible
by both 4 and 6, but it is not divisible by (4*6)=24, since 4 and 6 are
not co- primes.
Progression:
A succession of numbers formed and arranged in a definite order according to
certain definite rule, is called a progression.
1. Arithmetic Progression
(A.P.):
If each term of a progression differs from its preceding term
by a constant, then such a progression
is called an arithmetical progression.
This constant difference is called the common difference of
the A.P.
An A.P. with first term a and common difference d is given
bya,(a+d),(a+2d),(a+3d),.....
The nth term of this A.P. is given by Tn=a(n−1)d.
The sum of n terms of this A.P. Sn=n/2(2a+(n-1)d).
Some Important Results:
(i) (1+2+3+….+n)=
(ii) (l2+22+32+...+n2)=
n(n+1)(2n+1)
|
6
|
(iii) (13+23+33+...+n3)=n2(n+1)2
2. Geometrical
Progression (G.P.):
A progression of numbers in
which every term bears a constant ratio with its preceding
term, is called a geometrical progression. The constant ratio is called the
common ratio of the G.P.
A G.P. with first term a and common ratio r is :a,ar,ar2,…..
In this G.P.nth term, Tn=arn−1
sum of n terms, Sn=ax [(r^n-1)-1]/r-1]
when r<1
1.
Ratio:
The ratio of two quantities a and b in
the same units, is the fraction 
and
we write it as a : b.
and
we write it as a : b.
In the ratio a : b, we
call a as the first term or antecedent and
b, the second term or consequent.
Eg. The ratio 5 : 9 represents
|
5
|
with antecedent = 5, consequent = 9.
|
9
|
Rule: The
multiplication or division of each term of a ratio by the same non-zero number
does not affect the ratio.
Eg. 4 : 5 = 8 : 10 = 12 : 15. Also, 4 : 6 = 2 : 3.
2.
Proportion:
The equality of two ratios is called proportion.
If a : b = c : d,
we write a : b :: c : d and
we say that a, b, c, d are in proportion.
Here a and d are
called extremes, while b and c are
called mean terms.
Product of means = Product of extremes.
Thus, a : b :: c : d 
(b x c)
= (a x d).
(b x c)
= (a x d).
3.
Fourth Proportional:
If a : b = c : d,
then d is called the fourth proportional to a, b, c.
Third Proportional:
a : b = c : d,
then c is called the third proportion to a and b.
Mean Proportional:
Mean proportional between a and b is ab.
4.
Comparison of Ratios:
We say that (a : b)
> (c : d)
|
a
|
>
|
c
|
.
|
b
|
d
|
5.
Compounded Ratio:
6.
The
compounded ratio of the ratios: (a : b), (c : d),
(e : f) is (ace : bdf).
7.
Duplicate Ratios:
Duplicate ratio of (a : b) is (a2 : b2).
Sub-duplicate ratio of (a : b) is (a : b).
Triplicate ratio of (a : b) is (a3 : b3).
Sub-triplicate ratio of (a : b) is (a1/3 : b1/3).
If
|
a
|
=
|
c
|
, then
|
a + b
|
=
|
c + d
|
. [componendo and
dividendo]
|
b
|
d
|
a - b
|
c - d
|
8.
Variations:
We say that x is directly proportional
to y, if x = ky for some constant k and
we write, x 
y.
y.
We say that x is inversely proportional
to y, if xy = k for some constant k and
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