Number System: Basic Number Theory, arithmetic progression, Geometric progression
Posted on : 14-02-2019 Posted by : Admin

The following are the important points of basic number theory. Each of these is very important for solving simple to complex mathematical equations.

• Square of every even number is an even number.

Example: 2×2=4, 6×6=36...

All the products are even.

• Square of every odd number is an odd number.

Example: 3×3=9, 5×5=25...

All the products are odd.

• A number obtained by squaring a number doesn’t have 2, 3, 7 or 8 at its units place.

Example: 1×1=1, 2×2=4, 3×3=9…

Here we can observe that no number has 2, 3, 7 or 8 in units place.

• Sum of first n natural numbers = $\frac{\mathbf{n}\mathbf{(}\mathbf{n}\mathbf{+}\mathbf{1}\mathbf{)}}{\mathbf{2}}$
  

Example: Let us consider the sum of first 5 natural numbers i.e. n=5.

Sum=1+2+3+4+5=15.

By applying n=5 in the formula we get,

$\frac{5(5+1)}{2}=\frac{30}{2}=15$

• Sum of first odd numbers = n².

Example: Sum of first 4 odd numbers is 1 + 3 + 5 + 7 =16. By applying n = 4 in the formula. Sum = 4² = 16

• Sum of first n even numbers = n(n+1)

Example: Sum of first 5 even numbers is 2 + 4 + 6 + 8 + 10 = 30. By applying n = 5 in the formula. Sum = 5 (5+1) = 5 × 6 = 30

• Sum of squares of first n natural numbers = $\frac{\mathbf{n}\mathbf{(}\mathbf{n}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{ }\mathbf{(}\mathbf{2}\mathbf{n}\mathbf{+}\mathbf{1}\mathbf{)}}{\mathbf{6}}$

Example: Sum of squares of first 4 natural numbers is 1² + 2² + 3² + 4² = 1 + 4 + 9 + 16 = 30.

By applying n = 4 in the formula we get,

$$\begin{gathered} \frac{4(4+1) \left(2\right(4)+1)}{6} \\ =\frac{\left(20\right) \left(9\right)}{6}=30 \end{gathered}$$

• Sum of cubes of first n natural numbers = ${\left[\frac{{\displaystyle \mathbf{n}\mathbf{(}\mathbf{n}\mathbf{+}\mathbf{1}\mathbf{)}}}{{\displaystyle \mathbf{2}}}\right]}^{\mathbf{2}}$
  

Example: Sum of cubes of first 3 natural numbers is 1³ + 2³ + 3³ = 36

By applying n = 3 in the formula we get,

$$\begin{gathered} {\left[\frac{{\displaystyle 3(3+1)}}{{\displaystyle 2}}\right]}^2 \\ ={\left[\frac{{\displaystyle 12}}{{\displaystyle 2}}\right]}^2 = {\left[6\right]}^2 = 36 \end{gathered}$$

• Every prime number greater than 3 can be written in the form of (6k+1) or (6k-1) where k is an integer.

Example: Assume the value of k=1, we get 6(1) +1, 6(1) -1=7, 5.

There are 15 prime numbers between 1 and 50 and 10 prime numbers between 50 and 100.

• If p divides q and r, then p divides their sum and difference also.

Example: 4 divides 12 and 20, then 20+12=32 and 20-12=8 are also divisible by 4.

• For any natural number n, (n³-n) is divisible by 6.

Example: Applying n=2 in the equation, 2³-2=6 is divisible by 6.

• The product of three consecutive natural numbers is always divisible by 6.

Example: Let us take the product 7×8×9=56×9, which is divisible by 6.

• (x^m–a^m) is divisible by (x-a) for all values of m.

Example: Let us take m=2, x²-a² can be written as (x + a) (x - a) is divisible by (x-a)

• (x^m – a^m) is divisible by (x + a) for even values of m.

Example: Let us take m=2, x²-a² can be written as (x + a) (x – a) is divisible by(x + a)

• (x^m + a^m) is divisible by (x + a) for odd values of m.

Example: Let us take m=3, x³+a³ can be written as (x + a) (x² – ax + a²) is divisible by (x + a)

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