1. Number Theory

Number theory questions cover a wide array of concepts that range from simple questions on LCM, HCF to questions on factors, remainders, number of factors etc. 4GMAT's GMAT Math Lesson Book in this hard math topic covers the following concepts in Number Theory.

1. Definition of types of numbers from Natural Number to Real numbers, complex numbers and their conjugates.
2. Irrational numbers, surds, binomial quadratic surds and their conjugates.
3. Method to determine if a given number is a prime number.
4. Concepts and examples of co-prime numbers.
5. Tests of divisibility including shortcuts for divisibility of numbers such as '7' and '13'.
6. Concepts and word problem examples to elucidate LCM and HCF.
7. Fractions - Proper fractions, improper fractions, LCM and HCF of fractions.
8. 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.
9. Properties of remainders and quotients of division of large numbers such as 371.
10. Remainder theorem and its application with examples, which is considered to be a difficult topic with tough questions is explained with examples.
11. Factorials and highest power of a number contained in a factorial.
12. Properties of unit digits of numbers raised to large powers. For example unit digit of 3231179.
13. Conversion of numbers from one base to another and back to decimal notation.
14. 19 illustrative examples to explain the concepts; 34 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.

Here is an example of a typical solved example in this chapter.
Sample Question
Question
A green light flashes thrice in a minute and a red light flashes 4 times in 2 minutes. If the two lights start flashing together, how often will they flash together in an hour?

Explanatory Answer
The green light flashes thrice in a minute.
Therefore, the green light will flash once every 20 seconds.

The red light flashes 4 times in 2 minutes
Therefore, the red light will flash once every 30 seconds.

The green light will flash at multiples of 20 seconds. i.e., 20, 40, 60, 80 ... seconds.

The red light will flash at multiples of 30 seconds. i.e., 30, 60, 90, 120 ... seconds.

Hence, the two lights will flash together at time intervals that are common multiples to both 20 and 30.

The Least Common Multiple (LCM) of 20 and 30 = 2*3*10 = 60.
i.e., the two lights will flash together every 60 seconds or 1 minute.
Therefore, they will flash together 60 times in an hour.