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# Arithmetic : Properties of Integers and Numbers

 Preparation Just what you need to know !

Prime Numbers and Composite Numbers

Any positive integer can be expressed as the product of 1 and itself.
For example, 17 = 1 × 17; so, 1 and 17 are factors (divisors) of 17.

A prime number is a positive integer that has exactly two different positive factors (1 and itself).
Thus, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47,
53, 59, 61, 67, 71, 73, 79, 83, 89 and 97 are the prime numbers upto 100.

A composite number is a positive integer that has more than two different positive factors.
Except 1, any number which is not a prime number is a composite number.
The number 1 is neither prime nor composite because it has only one positive factor.

MUST-KNOW : Any composite number can be uniquely expressed as a product of prime factors.

To write a number as a product of primes, first divide the number by 2 as many times as possible, next divide the result by 3 as many times as possible, and then continue this procedure of dividing by the other successive prime numbers 5, 7, 11, 13 and so on, until all the factors are primes.
For example to express 84 as a product of prime numbers, note that 84 = 2 × 42 = 2 × 2 × 21 = 2 × 2 × 3 × 7.

Example
In how many different ways can 90 apples be arranged in rows on a table if there must be more than two rows, and more than two apples in each row. Each row must have the same number of apples. Note that m rows of n apples each is a different arrangement from n rows of m apples each.
Solution. Number of rows × Number of apples per row = Total number of apples = 90
So, we need two integers, each of which is greater than 2 and whose product is 90.
It is convenient to express 90 as a product of primes.
Now, 90 = 2 × 45 = 2 × 3 × 15 = 2 × 3 × 3 × 5.
The possible ways to express 90 (excluding 2 × 45 and 45 × 2) are 3 × 30, 5 × 18, 6 × 15, 9 × 10, 10 × 9, 15 × 6, 18 × 5, and 30 × 3.
So, there are 8 different ways.

GMAT Math Review - Arithmetic : Index for Integers & Numbers

10 more pages in GMAT Math Review

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