The best way to study mathematics is to find its applications in different subjects and fields of education and society. The vital thing to learn in mathematics is the formula creation method. How can any pattern be converted into a mathematical formula?
Let’s understand this through two examples.
First example: What are the rules for the divisibility of any number by 2? The last digit of that number must be even, meaning it should be divisible by 2. Now, what are the rules for the divisibility of any number by 4? The last two digits of that number must be divisible by 4. Next, what are the rules for the divisibility of any number by 8? The last three digits of that number must be divisible by 8. Now, just carefully observe the hidden pattern here. 4 = 2² and 8 = 2³. By looking at this, it becomes clear why the last 2 digits are checked for divisibility by 4 and the last 3 digits are checked for divisibility by 8.
Second example: If any number is divided by 9, what will be the remainder? Can this be known without actually dividing the number by 9? The answer is “YES”. Take the number you want to divide by 9 and add all its digits together. Then, add the digits of the resultant sum. Repeat this process until the result is a single digit. The final single-digit resultant will be the remainder that would have been obtained if the original number were divided by 9.
Let’s check it; If 5761 is divided by 9, what will be the remainder? Instead of using the division method, we’ll apply our method:
5 + 7 + 6 + 1 = 19
1 + 9 = 10
1 + 0 = 1
Even if we perform the division properly, the remainder will still be 1. Let’s try another number, 382946
3 + 8 + 2 + 9 + 4 + 6 = 32
3 + 2 = 5
Here, even if the order of the digits is changed, the remainder when divided by 9 will remain the same. This method can be tested with any number, and the formula is universally applicable.
Now, the question that arises here is, how is this formula created?
This formula is created by combining two different concepts. The first concept is the rule of divisibility by 9, which states that if the sum of all the digits of a number is divisible by 9, then the number itself is divisible by 9. The second concept is that 9 is the largest single-digit number. Therefore, if the sum of all the digits of a number is reduced to a single-digit number, we will get a number that is either equal to 9 or smaller than 9. This is why, even if the order of the digits changes, the same rules for checking divisibility by 9 will still be applicable.
To combine two different concepts, the most important thing required is observation, which can only be achieved through continuous practice.
Good observation requires a lot of practice, and practice comes from continuously solving questions. That is why, on this website, you will find numerous practice questions related to every topic in mathematics, with detailed explanations of solutions. Through these, people can become experts in mathematics. Additionally, these practice questions will be very useful for various competitive exams like IITJEE, GATE, CAT, GMAT, etc. So, find questions according to your topic and practice as much as you want.
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