Solving Number Theory Problems

Introduction of solving number theory problems
Number theory is the theory about numbers which is called "the queen of mathematics" by the legendary mathematician Carl Friedrich Gauss, number theory is one of the oldest and largest branches of pure mathematics. The number theory delves deep into the structure and nature of numbers, and explores the remarkable, often beautiful relationships among them.

The number theory has many different types of numbers:

Solving Natural numbers problemssolving Prime numbers problems
solving Integers problems
solving Algebraic numbers problems
solving imaginary numbers problems
solving transcendental numbers problems


Example problems for solving number theory


Example: 1

Find the values of consecutive numbers where the sum of the two numbers is 161.

Solution:

Let as assume the two consecutive numbers be x, x+1.

Where the sum is 161 so,

x + x + 1 = 161

2x + 1 = 161

2x = 160

x = 80

Therefore, x + 1 = 80 + 1 = 81

So the consecutive numbers is 80 and 81.

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Example: 2

Find all integers n such that n − 18 and n + 18 are both perfect Squares.

Solution:            Let as assume n −18 = a2 and n +18 = b2.

Then b2 −a2 = 36, so (b−a) (b+ a) = 22 .32.

Because b−a and b+a are of the same parity,

So, the following possibilities are:  b − a = 2, b + a = 18, yielding b = 10, a = 8,

And b − a = 6, b+a = 6, yielding a = 0, b = 6.

Hence the integers with this property are n = 46 and n = 18.


Example :3


Find the values of consecutive numbers where the product of the two numbers is 121.

Solution:-

Let as assume the two consecutive numbers be x, x +1.

Where the product is 121 so,

x * (x + 1) = 121

x^2 + x = 121

x^2 + x – 121 = 0

x^2 + 12x – 11x – 121 = 0

(x + 12) (x - 11) = 0

(x + 12) = 0 (or) (x - 11) = 0

x = -12 (or) x = 11

-12 is not possible to get 121

Therefore we take x = 11

So, x + 1 = 11 + 1 = 12

The consecutive numbers is 11 and 12.