Number System

Basic Formulae

1. ${\left(a+b\right)}^2=a^2+b^2+2ab$                     

2. ${\left(a-b\right)}^2=a^2+b^2-2ab$
3. ${\left(a+b\right)}^2-{\left(a-b\right)}^2=4ab$                       
4. ${\left(a+b\right)}^2+{\left(a-b\right)}^2=2\left(a^2+b^2\right)$
5. $\left(a^2–b^2\right)=\left(a+b\right)\left(a-b\right)$                     
6. ${\left(a+b+c\right)}^2=a^2+b^2+c^2+2\left(ab+bc+ca\right)$
7. $\left(a^3+b^3\right)=\left(a+b\right)\left(a^2-ab+b^2\right)$   
8. $\left(a^3–b^3\right)=\left(a-b\right)\left(a^2+ab+b^2\right)$
9. $\left(a^3+b^3+c^3-3abc\right)=\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ca\right)$
10. If $a+b+c=0$, then $a^3+b^3+c^3=3abc$.

Types of Numbers:

I. Natural Numbers:

Counting numbers $1,2,3,4,5,\dots \dots$ 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., $\dots ,-3,-2,-1,0,1,2,3,\dots ..$ together form the set of integers.
(i) Positive Integers: $1,2,3,4,\dots ..$ is the set of all positive integers.
(ii) Negative Integers: $-1,-2,-3,\dots ..$ is the set of all negative integers.
(iii) Non-Positive and Non-Negative Integers: 0 is neither positive nor negative. 
So,  $0,1,2,3,\dots .$  represents  the  set  of  non-negative  integers, 
while $0,-1,-2,-3,\dots ..$ 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 $\frac{p}{k}=q+\frac{r}{k}$
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 $10,000n+57=1,000\left(10n\right)+57$
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).
Method II (Alternative form)
Since  the  remainder  is  57  when  the  number  is  divided  by  10,000,  the  number  can  be expressed  as $10,000n+57$. Dividing this number by 1,000 yields
$\frac{10,000n+57}{1000}=\frac{10,000n}{1000}+\frac{57}{1000}=10n+\frac{57}{1000}$
Hence, the remainder is 57 (by the alternative form  $\frac{p}{k}=q+\frac{r}{k}$ ), and the answer is (D).

Even, Odd Numbers:

A number n is even if the remainder is zero when n is divided by $2:n=2z+0$, or $n=2z$.
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 should memorize them:
$\text{even * even = even}$
$\text{odd * odd = odd}$
$\text{even * odd = even}$
$\text{even + even = even}$
$\text{odd + odd = even}$
$\text{even + odd = odd}$

---

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:

$\left(n+1\right)\left(n+3\right)$ is odd only when both $\left(n+1\right)$ and $\left(n+3\right)$ are odd. This is possible only when n is even.
Hence, n = 2m, where m is a positive integer. Then,
$\left(n+2\right)\left(n+4\right)=\left(2m+2\right)\left(2m+4\right)=2\left(m+1\right)2\left(m+2\right)=4\left(m+1\right)\left(m+2\right)=$
$\text{4 * (product of two consecutive positive integers, one which must be even) =}$ $\text{4 * (an even number), and this equals a number that is at least a multiple of 8}$
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 $=\left(5+9+2+4+8+2\right)=30$, which is divisible by 3.
But, 864329 is not divisible by 3, since sum of its digits $=\left(8+6+4+3+2+9\right)=32$, 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$=\left(3+5+2+5+6\right)=21$, 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 $=\left(6+0+7+3+2\right)=18$, which is divisible by 9.
But, 68956 is not divisible by 9, since sum of digits $=\left(6+8+9+5+6\right)=34$, 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 $\text{:(sum of digits at odd places) - (sum of digits at even places) =}$
$=\left(8+7+3+4\right)-\left(1+2+8\right)=11$, 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 $=\left(3+4+6+3+2\right)=18$, 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 $\left(4*6\right)=24$, 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 by $a,\left(a+d\right),\left(a+2d\right),\left(a+3d\right),.....$
The nth term of this A.P. is given by $T_n=a\left(n-1\right)d$.
The sum of n terms of this A.P. $S_n=\left(\frac{n}{2}\right)\left[2a+\left(n-1\right)d\right]=\left(\frac{n}{2}\right)*\left(\text{first term + last term}\right).$

Some Important Results:

(i) $\left(1+2+3+\dots .+n\right)=\frac{n\left(n+1\right)}{2}$
(ii) $\left(l^2+2^2+3^2+...+n^2\right)=\frac{n\left(n+1\right)\left(2n+1\right)}{6}$
(iii) $\left(1^3+2^3+3^3+...+n^3\right)=n^2{\left(n+1\right)}^2$

---

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 $:a,ar,a{r}^2,\dots ..$
In this G.P.$n^{th}\text{term,}{T}_n=a{r}^{n-1}$
sum of  n terms, $S_n=\frac{a\left(1-r^n\right)}{\left(1-r\right)}$  when $r<1$

IBPS PO CWE 2013 Admit Card Download

The Call Letter / Admit Card for IBPS PO / MT CWE III is now available. As you all know that this exam is going to take place shortly. You can download your call letter / Admit card by visiting the link provided here. You have to fill in information as required and thereafter you can get your Call Letter.





http://ibpsreg.sifyitest.com/cweposep13/login.php?appid=qassdieica13ndwe4alq

Largest banks in the World ranked by Total Assets in –2013

(Total assets, US$bn & as per Balance
sheet on 31.03.2013)
1 Industrial & Commercial Bank of
China (ICBC) (China) — 2,953.85
2 HSBC Holdings (UK) — 2,681.36
3 Deutsche Bank (Germany) —
2,597.36
4 Credit Agricole Group (France) —
2,582.42
5 BNP Paribas (France) —
2,507.96(data by facebook/cnaonweb
6 Mitsubishi UFJ Financial Group
(Japan) 2,486.31
7 Barclays PLC (UK) — 2,414.78
8 JPMorgan Chase & Co (USA) —
2,389.35
9 China Construction Bank
Corporation (China) — 2,361.60
10 Japan Post Bank (Japan) —
2,118.84(data by facebook/cnaonweb
11 Agricultural Bank of China (China)
— 2,295.80
12 Bank of America (USA) — 2,174.61
13 Bank of China (China) — 2,130.82
14 Royal Bank of Scotland Group (UK)
— UK 1,979.14
15 Citigroup Inc (USA) —
1,881.73(data by facebook/cnaonweb
16 Mizuho Financial Group (Japan) —
1,881.03
17 Banco Santander (Spain) —
1,637.74
18 Societe Generale (France) —
1,592.51
19 Sumitomo Mitsui Financial Group
(Japan) — 1,576.58
20 ING Group (Netherlands) —
1,508.71

Data Sufficiency

NEARLY 20+ QUESTION WILL BE ASKED IN SBI AND OTHER PO EXAMS

This is also one interesting section of reasoning paper. Mastering this section is not very difficult but meanwhile may not be a cake walk too.

Example 1. In which year was Rahul born ?

Statements: Rahul at present is 25 years younger to his mother.

Rahul's brother, who was born in 1964, is 35 years younger to his mother.

I alone is sufficient while II alone is not sufficient

II alone is sufficient while I alone is not sufficient

Either I or II is sufficient

Neither I nor II is sufficient

Both I and II are sufficient

Solution: Take statement one & try solving but alone statement 1 will not be sufficient. However

statement 2 is also not sufficient alone, but taking both of the statement together question can be

solved. Hence answer is e.

Example 2. What will be the total weight of 10 poles, each of the same weight ?

Statements: One-fourth of the weight of each pole is 5 kg.

The total weight of three poles is 20 kilograms more than the total weight of two poles.

I alone is sufficient while II alone is not sufficient

II alone is sufficient while I alone is not sufficient

Either I or II is sufficient

Neither I nor II is sufficient

Both I and II are sufficient

Solution: From statement 1. W/4 = 5 therefore W=20 Kg. So total weight = 200 Kg.

From statement 2. 3W= 20+ 2W, W=20. So total weight is 200 Kg.

7

Therefore c is the correct answer. To understand various patterns you should practice well & try solving different type of questions.

GK QUESTIONS IN RRB

.1World Water Day – March 22
2. Argentina Capital - Buenos Aires 3.
Australia Currency - Dollar
4. First Indian Women to get Olympic
Medal –
Karnam Malleshwari
5. English Vinglish Movie Director –
Gauri
Shinde
6. International Development
Association HQ – Washington
7. Gnanpit Award Telugu Literature
2013 –
Ravauri Bharadwaja
8. Pride and Prejudice Book Author –
Jane
Austen
9. Fiat Cars Which Country – Italy 10.
Wriddhiman Saha related to – Cricket
11. Highest Literacy State (2011
Census) –
Kerala
12. Abdul Rahim Rather Minister of
state –
Jammu & Kashmir
13. Khajuraho – Madhya Pradesh 14.
Vinton Cerf – Father of Internet
15. Swaythling Cup for Men, Corbillon
Cup for
Women Related to Which Game –
Table Tennis
16. Difference Between Repo &
Reverse Repo
Rate – Interest Rate Corridor
17. Annapurna Scheme – Distribution
of Food Grains
18. JNNURM Started in Year – 2005
19. Current Chief Justice –
Sathasivam
20. Ranbaxy related to -
Pharmaceutical
21. Bulk Payments – ECS
22. A question about GDP – Measure
of Total Flow of Goods & Services
23. To Classify as NPA Number of
Required
Days - 90 Days
24. Number of Digits in IFSC Code - 7
25. Image Based Cheque Clearing -
Cheque
Truncation System 26. 14th Finance
commission Headed By – YV
Reddy
27. CASA Ratio – Share of Current and
Savings
Account
28. Situation where an inflation rate
is high, the
economic growth rate slows down,
and unemployment remains steadily
high –
Stagflation
29. Banks Capital Infusion – 14,000
crores by
May 2014
30. Safest Form of Deposit – Fixed
Deposit
31. Converting assets into marketable
securities & generating cash flows –
securitisation
32. If NRI wants to invest in stock
exchange,
which account should b opened..? –
NRO Demat
Account
33. Company Buying its own shares –
Share Buy Back
34. Dividend Financial Ratio –
35. To improve improve condition of
currency
notes RBI's launched which pilot
project –
Plastic Currency
36. Legal term used to represent
insurance before issue of Policy
Document – Cover Note

Statement & Conclusions shortcuts

important topic in ibps po.from this you can expect 5 question surely.be ready prepare well

This section can be really difficult and actually helps exercising your brain cells. Apart from checking your problem solving skills this also checks your ability of choosing the correct questions to answer, because there will be some questions placed deliberately to trap you & will absorb your time.

To believe it or not but there no direct shortcuts or tricks to solve these questions but you can actually train your brain by practicing hard because that will help you learn various patterns. Anyways don’t worry that much, let’s learns few patterns & understand how this section can also be conquered.

Generally the questions in this section are of the following type

Type 1: Statement: In a cricket match, the total runs made by the team were 200. Out of these 160

runs were made by spinners.

Conclusions:

80% of the team consists of spinners.

The opening batsmen were spinners.

Only conclusion I follows

Only conclusion II follows

Either I or II follows

Neither I nor II follows

Both I and II follow

Solution: Now if we calculate (160/200) *100= 80%. Which simplifies the statement into “80%

runs were made by spinners” but that doesn’t conclude that “80% of the team consists of spinners”. Conclusion 1 is simply to confuse your choice. However Conclusion 2 can be easily

removed as given statement doesn’t talk anything about opening batsman.

Type 2: Choose the conclusion which logically follows given statement

8

Statement: Soldiers serve their country.

Men generally serve their country.

Those who serve their country are soldiers.

Some men who are soldiers serve their country.

Women do not serve their country because they are not soldiers.

Solution: Conclusion a b & d can be easily removed as statement doesn’t conclude that only

en serve their country or only soldiers can serve their country. However conclusion b is quite lose but doesn’t include that others can also serve their country. Hence conclusion c logically  follows the given statement because this simply leaves the space that women & others can also

serve country.