Number Properties and Number Theory is a topic from which you should expect 4 to 7 questions in the GMAT quant section - in both variants viz., problem solving and data sufficiency. This is a very important topic.

The GMAT Syllabus in Number properties, number systems, and number theory include a wide array of concepts that range from simple questions on LCM, HCF, to questions on factors, remainders, remainder theory, factorial, unit digits, number of factors, sum of factors, and polynomial division rules . Wizako's GMAT Math Lesson Book in this hard math topic covers the following concepts in Number Properties & Number Theory.

- Number Systems: Definition of types of numbers from natural numbers to real numbers, complex numbers and their conjugates.
- Number Systems: Properties and operations on irrational numbers, surds, binomial quadratic surds and their conjugates.
- Number Systems: Method to determine whether a given number is a prime number.
- Number Systems: Concepts and examples of co-prime numbers.
- Tests of divisibility including short cuts for test of divisibility of numbers such as '7' and '13'.
- Concepts including how to compute, what it means and word problem examples to elucidate LCM and HCF.
- Fractions - Proper fractions, improper fractions, LCM and HCF of fractions.
- Factors and how to find the number of factors of an integer?
- How to find the sum of all positive factors of a natural number?
- How to find the product of all positive factors of a natural number?
- Perfect squares and number of factors.
- Number of ways of expressing an integer as a product of two of its factors.
- Euclid's division algorithm.
- Properties of remainders and quotients of division of large numbers such as 3
^{71}. - Remainder theorem and its application with examples, which is considered a difficult topic with tough questions is explained with examples.
- Factorials and highest power of a number contained in a factorial.
- Number of trailing zeroes in the value of a factorial.
- Properties of unit digits of numbers raised to large powers. For example unit digit of 323
^{1179}. - 21 illustrative examples to explain the concepts; 31 solved examples (with short cuts wherever applicable) to acquaint you with as many different questions as possible; 26 exercise problems with answer key and explanatory answers to provide you with practice and a timed multiple choice GMAT level test with 75 questions. Explanatory answers and answer key are provided for the test.

Here is a typical solved example in Wizako's GMAT Book from this chapter

A green light flashes thrice in a minute and a red light flashes 4 times in 2 minutes at regular intervals. If the two lights start flashing together, how often will they flash together in an hour?

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.

Starting from the 0th second, the green light will flash at multiples of 20 seconds. i.e., at 0, 20, 40, 60, 80 ... seconds.

Starting from the 0th second, the red light will flash at multiples of 30 seconds. i.e., at 0, 30, 60, 90, 120 ... seconds.

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

The smallest and first common multiple of 20 and 30 is the LCM of the two numbers.

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.

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