Number Theory, Number Systems : CAT Quant
Since CAT 2001, this is an oft-repeated topic in CAT
Number Theory / Number systems is a topic from which you can expect anywhere between 4 to 10 marks worth of questions in the quant section of CAT. This topic will definitely feature amongst the A category topics if one were to do an ABC analysis of topics that figure in CAT Math.
Ascent's Arithmetic Refresher covers this topic in depth. The chapter starts from absolute basic and covers advanced topics in Number Theory. The following areas are covered in this topic in the refresher material.
- Definition of different types of numbers from Natural Number to Real numbers, including irrational numbers, surds, binomial quadratic surds and their conjugates, complex numbers and their conjugates.
- Prime numbers and method to determine if a given number is a prime number. Co-prime numbers, applications and examples.
- Tests of divisibility including shortcuts for divisibility of numbers such as '7' and '13'.
- Concepts and solved examples to elucidate the concept of LCM and HCF.
- Fractions - Proper fractions, improper fractions, LCM and HCF of fractions.
- Factors and method to find the number of factors for a given integer. Method to find the sum of all positive factors of a natural number.
- Properties of remainders and quotients of division of large numbers for e.g., 371 by 8.
- Remainder theorem and its application with examples - an oft repeated question in CAT since CAT 2001.
- Factorials explained with examples. Highest power of a number contained in a factorial.
- Properties of unit digits of numbers raised to large powers. For example unit digit of 14328971179.
- Conversion of numbers from one base to another and back to decimal notation.
- Successive division of numbers and remainders of such successive division
The chapter contains
- Over 15 illustrative examples to explain the concepts
- Around 35 solved examples (with shortcuts wherever applicable) to acquaint you with as many different questions as possible
- 30 exercise problems with answer key and explanatory answers to provide you with practice and
- An objective type speed test with around 60 questions. Explanatory answers and answer key are provided for the speed test
Sample Question in Number Theory
This question appeared as a 2-Marks question in CAT 2005
What is the righmost non-zero digit of the number 302720?
(1) 1
(2) 3
(3) 7
(4) 9
Correct Answer: Choice (1).
Explanatory Answer
As the given number is a multiple of 10, the last 2720 digits will be 0s.
The right most non-zero digit of the number will be the unit digit of the number 32720.
The unit digit of the powers of "3" follow a cyclicity of "4"
i.e., The units digit of 31 is 3
The units digit of 32 is 9
The units digit of 33 is 7
The units digit of 34 is 1. Then the cyclicity sets in.
The units digits of 35 is 3.
The units digits of 36 is 9 and so on.
In 32720 as 2720 is a multiple of "4", the number will have the same units digit as 34 which is "1".
Hence, the rightmost non-zero digit of the number is "1".
Click here for additional sample questions in Number Theory
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