Numbers Questions Answers Formulas, Tips and Tricks
1. Type of numbers
Natural Numbers: Counting 1,2,3,4,5... are called natural numbers.
Whole Numbers: All counting numbers together with zero are whole numbers. So 0 is only a whole number which is not a natural number
Integers: All counting numbers, 0 and negative number together form the set of integers.
i.e. [....,-2,-1,0,1,2,.....] are integers.
Prime numbers: A number greater than 1 is called a prime number if it has exactly two factors, namely 1 and the number itself.
Composite numbers: Numbers greater than 1 which are not prime are called composite numbers.
Co-primes: Two numbers are said to be co primes if their H.C.F is 1. example: (2,3),(8,11) etc2. Methods for division
Divisibility by 2: A number is divisible by 2, If its unit digit is any of 0,2,4,6,8
Example: 3436, 12, 4570 etc
Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
Example: 324 as (3+2+4 = 9) so 9 is divisible by 3, hence 324 will be divisible by 3
Divisibility by 4: A number is divisible by 4 if the number formed by the last two digits is divisible by 4
Example: 2348 is divisible by 4 as the number formed by last two digits is 48 which is divisible by 4. So number 2348 is divisible by 4
Divisibility by 5: A number is divisible by 5 if its unit digit is either 0 or 5.
Divisibility by 6: A number is divisible by 6 if the number is divisible by both 2 and 3
Divisibility by 8: A number is divisible by 8 if the number formed by the last three digits is divisible by 8
Example: 2348360 is divisible by 8 as the number formed by last three digits is 360 which is divisible by 8. So number 2348360 is divisible by 8
Divisibility by 9: A number is divisible by 9, if the sum of its digits is divisible by 9.
Example: 13572 is divisible by 9 as sum of digits is (1+3+5+7+2 = 18), which is divisible by 9.
Divisibility by 10: A number is divisible by 10, if it ends with 0.
Divisibility by 11: A number is divisible by 11, if the difference of the sum of its digits at odd places and sum of digits at even places is either 0 or a number divisible by 11.
Example: 25795 is divisible by 11 as ((2+7+5)-(5+9) = 0)
Divisibility by 12: A number is divisible by 12 if it is divisible by both 4 and 3.
Divisibility by 14: A number is divisible by 12 if it is divisible by both 7 and 2.
Divisibility by 15: A number is divisible by 15 if it is divisible by both 5 and 3.
Divisibility by 16: A number is divisible by 16 if number formed by last 4 digits is divisible by 16.
Example: 886976 is divisible by 16 as last four digits 6976 are divisible by 16.
Divisibility by 24: A number is divisible by 24 if it is divisible by both 8 and 3.
Divisibility by 40: A number is divisible by 40 if it is divisible by both 8 and 5.
Divisibility by 80: A number is divisible by 80 if it is divisible by both 16 and 5.3. Multiplication methods
1. Multiplication by distributive law:
i. a * (b + c) = a * b + a * c
ii. a * (b - c) = a * b - a * c
2. Multiplication by \begin{aligned}5^n\end{aligned}:
When need to multiply with 5 power n , then we can
put n zeros to the right of multiplicand and divide the number so formed by\begin{aligned}2^n\end{aligned}
Example: 576 * 625
\begin{aligned}
= 576 * 5^4 \\
= \frac{5760000}{2^4}\\
= \frac{5760000}{16}\\
= 360000
\end{aligned}
4. Progression Formulas
A succession of numbers formed and arranged in a definite order according to certain definite rule, is called a progression.
We have two type of progressions
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 arithmetic progression. The constant is called as common difference of the Arithmetic Progression.
For example:
1, 3, 5, 7, 9...
this sequence is in Arithmetic Progression by common difference 2.
There are few important results for Arithmetic Progression, which you should be aware of.
Nth term in Arithmetic Progression(A.P.)= a + (n-1)d
Sum of n terms of this A.P.
\begin{aligned}
= \frac{n}{2}[2a+(n-1)d] \\
\text{or } \frac{n}{2}(\text{first term + last term})
\end{aligned}
Other Important Results:
\begin{aligned}
(1+2+3+..+n) = \frac{n(n+1)}{2} \\
(1^2+2^2+3^2+..+n^2) = \frac{n(n+1)(2n+1)}{6}\\
(1^3+2^3+3^3+..+n^3) = \frac{n^2(n+1)^2}{4}\\
\end{aligned}
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.
Example:
\begin{aligned}
a, ar,ar^2, ar^3,...
\end{aligned}
In Geometrical Progression(G.P.) nth term is
\begin{aligned}
= ar^{n-1}
\end{aligned}
In Geometrical Progression(G.P.) Sum of the n terms
\begin{aligned}
= \frac{a(1-r^n)}{1-r}
\end{aligned}
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