Number System is an all-time favourite with the CAT exam setters as well as the students. The topic is known for intriguing conceptual problems that test the best brains. For this topic, you need to use a combination of pure mathematical knowledge and logic skills. In fact, this is one topic that begins with the absolute basics of mathematics, that is, numbers. It explains to you the different types of numbers and then extends the application of this knowledge to a variety of domains.

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Number system is such a vast area that the syllabus for the same cannot be defined exactly. Keeping in mind the concepts from which questions can appear in the exam, the main topics for this area include **unit's digit, reminders, digital root, divisibility rules, integers and number of zeroes**. Besides these topics, questions are also based on **real numbers, prime numbers, natural numbers, and whole numbers**. In short, questions on number system may be based on basic definitions as well as advanced concepts.

- You should understand the basic definitions of numbers. Also, be careful when you attempt a question and identify the number type the question probes. There is a significant difference between real numbers, natural numbers, and integers. Can you recall the difference right now?
- Make sure you learn all prime numbers till 100.
- Advanced topics such as number of zeroes or the highest power, unit's digit, digital root and Euler number should be covered thoroughly. In fact, we have individual articles for these topics in the Quantitative Aptitude section.

- A sufficient number of questions of each of these areas must be solved to get the expertise in the area. Remember, when you practice questions in bulk for topics such as Unit Digit, you make sure that the process of solving that question is clear in your mind.

- Every number has the same unit’s digit at its fifth power as it has at its first power, thus the standard method that can be followed is to divide the power given by 4, find the remaining power and check the unit’s digit in that number. This short cut can be applied because you will always get the same unit’s digit as otherwise.
- For calculating number of zeroes at the end of factorial of a number, you should divide the number by 5, the quotient obtained is again divided by 5 and so on till the last quotient obtained is smaller than 5. The sum of all the quotients is the number of 5s, which then becomes the number of zeroes in the given number.
- The digital root of a number is the sum of the digits, over and over again, till it becomes a single digit number. For example, the digital root of 87984 will be 8 + 7 + 9 + 8 + 4 ⇒ 36 = 3 + 6 ⇒ 9.
- When the concept of Euler number is used and the dividend and divisor happen to be co-prime, the remainder questions become very easy.
- The product of 3 consecutive natural numbers is divisible by 6.
- The product of 3 consecutive natural numbers, the first of which is an even number is divisible by 24.
- The sum of a two-digit number and a number formed by reversing its digits is divisible by 11. E.g. 28 + 82 = 110, this is divisible by 11. At the same time, the difference between those numbers will be divisible by 9. e.g. 82 – 28 = 54, this is divisible by 9.
- ∑n = n(n+1)/2, ∑n is the sum of first n natural numbers.
- ∑n
^{2}= n(n+1)(n+2)/6, ∑n^{2}is the sum of first n perfect squares. - ∑n
^{3}= n^{2}(n+1)^{2}/4 = (∑n)^{2}, ∑n^{3}is the sum of first n perfect cubes. - x
^{n}+ y^{n}= (x + y) (x^{n-1}- x^{n-2}.y + x^{n-3}.y^{2}- ... +y^{n-1}) when n is odd. Therefore, when n is odd, x^{n}+ y^{n}is divisible by x + y. e.g. 3^{3}+ 2^{3}= 35 and is divisible by 5, which is (3 + 2). - x
^{n}- y^{n}= (x + y) (x^{n-1}- x^{n-2}.y + ... y^{n-1}) when n is even. Therefore, when n is even, x^{n}- y^{n}is divisible by x + y. e.g. 7^{2}- 3^{2}= 40, is divisible by 10, which is (7+3). - x
^{n}- y^{n}= (x - y) (x^{n-1}+ x^{n-2}.y + .... + y^{n-1}) for both odd and even n. Therefore, xn - yn is divisible by x - y. For example: 9^{4}- 2^{4}= 6545, is divisible by 7, which is (9 – 2).

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Example 4: What will be the number of zeroes at the end of the product of the first 100 natural numbers?

*Solution:* In this kind of questions, you need to find greatest power of 5, which can divide the product of the first 100 natural numbers. Remember, a multiple of 5 multiplied by any even number, gives you a zero. Now divide 100 by 5 and take 20 as quotient. Then divide 20 (the quotient) by 5 and get the new quotient 4, which further cannot be divided by 5. The sum of all such quotient gives you the greatest power of 5, which can divide that number. The sum is 24 and this is the number of zeroes at the end of the product of the first 100 natural numbers.