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Simplification:

TYPES OF NUMERALS:

1). Natural Numbers: These are counting numbers. For Ex. 1,2,3,4,5……

a) Natural numbers are denoted by ‘N’.

b)  All natural numbers are positive only.

c)  0 is not a natural number. The smallest natural number is ‘1’

2). Whole Numbers: Whole numbers having ‘0’ and natural numbers. Ex.0, 1, 2, 3, 4…….

a) Denoted by ‘W’

3) Integers: Negative numbers and whole numbers forms integers. Ex. -3, -2, -1, 0, 1, 2, 3….

a) Denoted by ‘I’

b) ‘0’ number is neither positive nor negative.

4) Even and Odd Numbers: The counting numbers which are divisible by 2 then those numbers called Even numbers otherwise Odd numbers.

a) Ex. For even numbers: 2, 4, 6, 8,10 ….

b) Ex. For odd numbers: 1, 3, 5, 7,9 ….

5) Prime Numbers: These numbers having only two factors ‘1’ and itself.

Ex. 2, 3, 5, 7, 11, 13 …. 2 is the even number which is prime and prime number is always greater than 1.

6) Rational and Irrational Numbers: Rational numbers are represented in the form of (a/b) where b is not equal to ‘0’, a and b are integers. Ex. (1/2), (2/5), (3/8)…

Irrational numbers cannot be represent in the form of (a/b). Ex. Square roof of 2

7) Composite Numbers: These numbers are not prime numbers and having at least one factor other than ‘1’ and itself.For Ex.  4, 8, 12…

Divisibility Rules:

1. Divisibility by 2: If the number is having either ‘0’ or even number as a last digit then that number is divisible by 2.

Example: 24, 40, 68, 122

2. Divisibility by 3: When sum of the digits of a number is divisible by 3, then the number is divisible by 3.

Example: 159     1 + 5 + 9 = 15     15 is divisible by 3.

3. Divisibility by 4: If the last two digits of a number is divisible by 4, then that number is divisible by 4.

Example: 124     24 is divisible by 4

If the number is having two or more zeroes at the end then also its divisible by 4.

Example: 1200   it’s having two zeroes at the end so its divisible by 4.

4. Divisibility by 5: If the number is having ‘0’ or ‘5’ at the end then it’s divisible by 5.

Example: 150, 205, 300

5. Divisibility by 6: If the number is divisible by 2 and 3then that number should divisible 6.

Example: 36, 54, 60

6. Divisibility by 7: when the difference between twice the digit at last place and the number formed by other digits is either ‘0’ or multiple of ‘7’.

Example:  147    14-(2*7) = 0        144 is divisible by 7

7. Divisibility by 8: If the number made by last three digits is divisible by 8, the number is divisible by 8.

Example: 24032                 032 is divisible 8

8. Divisibility by 9: It’s similar to divisible by 3 rule. When sum of the all the digits of a number is divisible by 9, then the number is divisible by 9.

Example: 4374                   4 + 3 + 7 + 4 = 18               18 is divisible by 9

9. Divisibility by 10: If the number is end with ‘0’ then it’s divisible by 10.

Example: 50, 110, 2000

10. Divisibility by 11: If the sum of digits at odd and even places are equal or differ by a number which is divisible by 11 then the number is divisible by 11.

Example: 216282              sum at even places(2+6+8) = 16;                               sum at odd places (1+2+2) = 5

Difference: (16-5) = 11. So 216282 is divisible by 11.