true

Find the best tutors and institutes for Class 10 Tuition

Find Best Class 10 Tuition

Please select a Category.

Please select a Locality.

No matching category found.

No matching Locality found.

Outside India?

Number Theory

Shreyan Saha
22 Jul 0 0

In number theory, we deal only with integers and their properties. Many of you may have heard about this branch of mathematics but may not know much about it. Here I am going to introduce you to some of the basic concepts of this branch of mathematics. So, let's begin :)

One of the primary concepts essential for learning number theory is the divisibility of integers. The first of this list is Euclid's division algorithm which states that:

Given integers a and b with b>0, there exist unique integers q and r such that a=qb+r.

Well, this is what we have learnt while learning division—considering an integer a and an integer b if we divide a by b. we get a quotient and a remainder. In the above theorem, q is the quotient and r is the remainder. But the exciting thing is that such a representation of an integer is always unique, which is quite apparent. 

Let's see how can we use this property to solve some interesting problems. But before that, you all must be knowing that an even integer N can be written as N=2k where k is also an integer. If N is odd, it can be represented as N=2k+1.

Problem 1: For n≥1 establish that the integer M=n(7n+5) is divisible by6, where n is an integer.

Solution: What do you think? Isn't it interesting that a number of this form will always be divisible by 6 irrespective of what value we put in place of n, if we place 1, it's true, same for 2, same for any n? But seeing the form, you cannot tell that the number will be divisible by 6.

So, let's see how we can approach the problem.

M=n(7n+5) =n(6n+ n+6-1)=6n+ n+6n -n = (6n3+ 6n) + (n-n). Let me tell you how this helps. We needed to prove that that M is divisible by 6. So, I decided to take out those terms which have 6 as a factor. Quite clearly  (6n3+ 6n)  will be divisible by 6. We just need to prove that (n-n)  will be divisible by 6. 

Let's factorise (n-n). (n-n)=n(n+1)(n-1) . Now suppose we have a number and we divide it by 3, the possible remainders are 0,1 and 2. So we can write any integer in the following forms: 3k,3k+1,3k+2  where k is an integer. Let's see what happens if we replace n with these forms.

If n=3k. So, (n-n)=3k(3k+1)(3k-1). So, (n-n) is divisible by 3.

If n=3k+1, (n-n)=(3k+1)(3k+2)3k. So, (n-n) is divisible by 3.

If n=3k+2, (n-n)=(3k+2)(3k+3)(3k+1)=3(3k+2)(k+1)(3k+3) NOTE: HERE WE TOOK 3 COOMON FROM 3K+3. WE WROTE 3K+3=3*(k+1).  So, (n-n) is divisible by 3.

So, for all integers,  (n-n) is divisible by 3.

We also know that an integer is either odd or even. So, n-2k if n is even or n=2k+1 if n is odd.

If n is even, (n-n)=2k(2k+1)(2k-1) .So, (n-n) is divisible by 2.

If n is odd, (n-n)=2k+1(2k+2)2k. So, (n-n) is divisible by 2.

So, (n-n)  is divisible by both 2 and 3. So, (n-n)  is divisible by 2*3=6.

Hence, we have proved that M is divisible by 6.

0 Dislike
Follow 2

Please Enter a comment

Submit

Other Lessons for You

Trigonometry
We try to understand what the physical meaning of trigonometry is? Trigonometry is nothing, but it is the relationship between the sides and angles of a right-angle triangle. For example - In a right...
S

Sameer Khan | 14 Aug

0 0
0

What is Co-prime Numbers?
We all know the prime numbers. Any number which is only divisible by '1' is called a prime number. Do you know what is coprime? First of all, it is a relative term, i.e. we can't say any number coprime...

Shubham Mishra | 13 Aug

0 1
1

Rotation of a Shape
* Rotation is nothing but rotating an object through an angle of 90 degrees, 180 degrees, 270 degrees either clockwise or anticlockwise. * For rotating 90 degrees clockwise and anticlockwise the steps...

Rasheedahamed F | 29 May

0 0
0

Mathematics
Vedic Mathematics Vedic Mathematics is a Mathematical elaboration of “sixteen simple Mathematical formulae from the Vedas”. As brought out by “Sri Bharati Krishna Tirthaji.” This...

Kavya K N | 14 May

0 0
0

Why am I learning Maths when I am not going to use it in everyday life?
A lot of students ask this question with a sense of boredom and servitude, "Why to study math if I can't use it ?". The answer to the question is pretty simple. You do use it in everyday life. Most of...

Looking for Class 10 Tuition ?

Find best Class 10 Tuition in your locality on UrbanPro.

Are you a Tutor or Training Institute?

Join UrbanPro Today to find students near you
Sponsored
X

Looking for Class 10 Tuition Classes?

Find best tutors for Class 10 Tuition Classes by posting a requirement.

  • Post a learning requirement
  • Get customized responses
  • Compare and select the best

Looking for Class 10 Tuition Classes?

Find best Class 10 Tuition Classes in your locality on UrbanPro

Post your learning requirement

UrbanPro.com is India's largest network of most trusted tutors and institutes. Over 25 lakh students rely on UrbanPro.com, to fulfill their learning requirements across 1,000+ categories. Using UrbanPro.com, parents, and students can compare multiple Tutors and Institutes and choose the one that best suits their requirements. More than 6.5 lakh verified Tutors and Institutes are helping millions of students every day and growing their tutoring business on UrbanPro.com. Whether you are looking for a tutor to learn mathematics, a German language trainer to brush up your German language skills or an institute to upgrade your IT skills, we have got the best selection of Tutors and Training Institutes for you. Read more