NUMBER SYSTEM

Sum of the first n odd numbers: N2

Sum of first n even numbers n(n+1)
Sum of the first n natural numbers [n(n+1)]/2
Sum of the squares of the first n natural numbers [n(n+1)(2n+1)]/6
Sum of the cubes of the first n natural numbers [n(n+1)/2]2

• (a + b)(a - b) = (a2 - b2)

• (a + b)2 = (a2 + b2 + 2ab)
• (a - b)2 = (a2 + b2 - 2ab)
• (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
• (a3 + b3) = (a + b)(a2 - ab + b2)
• (a3 - b3) = (a - b)(a2 + ab + b2)
• (a3 + b3 + c3 - 3abc) = (a + b + c)(a2 + b2 + c2 - ab - bc - ac)
• When a + b + c = 0, then a3 + b3 + c3 = 3abc
• (x3 + 1) = (x + 1)(x2 - x + 1)
• (a + b)2 + (a - b)2 = 2(a2 + b2)

Divisibility by
Divisibility Rule
number
A number that is even or a number whose last digit is an even number i.e.
Divisible by 2
0, 2, 4, 6, and 8.
Divisible by 3 The sum of all the digits of the number should be divisible by 3.
Number formed by the last two digits of the number should be divisible by
Divisible by 4
4 or should be 00.
Divisible by 5 Numbers having 0 or 5 as their ones place digit.
Divisible by 6 A number that is divisible by both 2 and 3.
Subtracting twice the last digit of the number from the remaining digits
Divisible by 7
gives a multiple of 7.
Number formed by the last three digits of the number should be divisible
Divisible by 8
by 8 or should be 000.
Divisible by 9 The sum of all the digits of the number should be divisible by 9.
Divisible by 10 Any number whose one's place digit is 0.
The difference of the sums of the alternative digits of a number is divisible
Divisible by 11
by 11.
Divisible by 12 A number that is divisible by both 3 and 4.
Multiply the last digit, i.e. unit digit by 9 of a number N and subtract it
Divisible by 13 from the rest of the number. If the outcome is divisible by 13 then the
number N is divisible by 13.
Cyclicity Of Numbers
1) Digits 0, 1, 5, and 6: Here, when each of these digits is raised to any power, the unit digit
of the final answer is the number itself.
2) Digits 4 and 9: Both of these two digits, 4 and 9, have a cyclicity of two different digits
as their unit digit.

4odd = 4: If 4 is raised to the power of an odd number, then the unit digit will be 4.

4even = 6: If 4 is raised to the power of an even number, then the unit digit will be 6.

9odd = 9: If 9 is raised to the power of an odd number, then the unit digit will be 9.

9even = 1: If 9 is raised to the power of an even number, then the unit digit will be 1.

3) Digits 2, 3, 7, and 8: These numbers have a cyclicity of four different numbers.

The cyclicity of 2 has 4 different numbers: 2, 4, 8, 6

The cyclicity of 3 has 4 different numbers: 3, 9, 7, 1.

The cyclicity of 7 has 4 different numbers: 7, 9, 3, 1.

The cyclicity of 8 has 4 different numbers: 8, 4, 2, 4.

PROBLEMS:
1. Which one of the following is not a prime number?
A.31
B. 61
C. 71
D.91

2. (112 x 54) = ?
A.67000
B. 70000
C. 76500
D.77200

3. what is the sum of smallest and highest number formed from the digits 6,1,4,2 and 5?
a. 52956
b. 52965
c. 77877
d. 77868
4. What least number must be added to 1056, so that the sum is completely divisible by 23 ?
A.2
 B. 3
C. 18
D.21
E. None of these

5. 1397 x 1397 = ?
A.1951609
B. 1981709
C. 18362619
D.2031719
E. None of these

6. The largest 4 digit number exactly divisible by 88 is:

A.9944
B. 9768
C. 9988
D.8888
E. None of these

7. What is the unit digit in {(6374)1793 x (625)317 x (341491)}?

A.0
B. 2
C. 3
D.5

8. The sum of first five prime numbers is:

A.11
B. 18
C. 26
D.28

9. If the number 517*324 is completely divisible by 3, then the smallest whole number in the
place of * will be:
A.0
B. 1
C. 2
D.None of these

11. Which one of the following numbers is exactly divisible by 11?

A.235641
B. 245642
C. 315624
D.415624

12. The sum of first 45 natural numbers is:

 A.1035
B. 1280
C. 2070
D.2140

13. Which of the following number is divisible by 24 ?

A.35718
B. 63810
C. 537804
D.3125736

14. 753 x 753 + 247 x 247 - 753 x 247

=?
753 x 753 x 753 + 247 x 247 x 247
1
A.
1000
1
B.
506
253
C.
500
D.None of these

15. The difference between the place value and the face value of 7 in the numeral 32675149
is
A.75142
B. 64851
C. 5149
D.69993
E. None of these

16. The difference between a positive proper fraction and its reciprocal is 9/20. The fraction
is:
3
A.
5
3
B.
10
4
C.
5
4
D.
3

17. On dividing a number by 56, we get 29 as remainder. On dividing the same number by 8,
what will be the remainder ?
A.4
B. 5
C. 6
D.7
 18. what will be the reminder when (6767+67) is divided by 68?
a. 1
b. 63
c. 66
d. 67

19. How many natural numbers are there between 23 and 100 which are exactly divisible
by 6 ?
A.8
B. 11
C. 12
D.13
E. None of these

20. A 3-digit number 4a3 is added to another 3-digit number 984 to give a 4-digit number
13b7, which is divisible by 11. Then, (a + b) = ?
A.10
B. 11
C. 12
D.15

21. If the product 4864 x 9 P 2 is divisible by 12, then the value of P is:
A.2
B. 5
C. 6
D.8
E. None of these

22. Which of the following numbers will completely divide (325 + 326 + 327 + 328) ?
A.11
B. 16
C. 25
D.30

23. What is the unit digit in(795 - 358)?

A.0
B. 4
C. 6
D.7
PRACTICE
1. If the number 481 * 673 is completely divisible by 9, then the smallest whole number in
place of * will be:
A.2
B. 5
C. 6
D.7
E. None of these

2. 107 x 107 + 93 x 93 = ?
A.19578
B. 19418
C. 20098
D.21908
E. None of these

3. On dividing a number by 5, we get 3 as remainder. What will the remainder when the
square of the this number is divided by 5 ?
A.0
B. 1
C. 2
D.4

4. How many 3-digit numbers are completely divisible 6 ?

A.149
B. 150
C. 151
D.166

5. What smallest number should be added to 4456 so that the sum is completely divisible by 6
?
A.4
B. 3
C. 2
D.1
E. None of these

6. 1 2 3
1 - + 1 - + 1 - + ... up to n terms = ?
n n n
HINT: AP
1
A. n
2
1
B. (n - 1)
2
1
C. n(n - 1)
2
 D.None of these

7. The sum of the two numbers is 12 and their product is 35. What is the sum of the
reciprocals of these numbers ?
12
A.
35
1
B.
35
35
C.
8
7
D.
32

8. (12 + 22 + 32 + ... + 102) = ?

A.330
B. 345
C. 365
D.385

9. The difference between the place value and the face value of 6 in the numeral 856973 is
A.973
B. 6973
C. 5994
D.None of these

10. What is the unit digit in (4137)754?

A.1
B. 3
C. 7
D.9