Prime Numbers + Divisibility

Let’s start this lesson with a basic lesson in divisibility.

Divisibility rules – you should memorize these!  A number is divisible by…

  • by 2 if the number is even or ends in 2,4,6,8 or 0 (this should be obvious)
  • by 3 if the sum of the digits is divisible by 3.   Example:  the number 43215 is divisible by 3 because 4+3+2+1+5 = 15 and 15 is divisible by 3.    The number is 3229 is not divisible by 3 because 3+2+2+9 = 16 and 16 is not divisible by 3.
  • by 4 if the last two digits are divisible by 4
  • by 5 if the last digit is a 5 or 0.
  • by 6 if a number is divisible by both 2 and 3
  • no easy rule for 7.
  • if the last three digits are divisible by 8
  • if the sum of the
  • digits is divisible by 9. (Example: 2349 is divisible by 9 because 2+3+4+9= 18 and 18 is divisible by 9)
  • by 10 if the last digit is 0.

Moving on to Prime Numbers
Prime numbers are pretty cool. They are the building blocks of numbers and they can tell us a lot about numbers. They also happen to be an important concept on GMAT math questions.

Facts about Prime Numbers

  • A prime number is a number that is divisible by 1 and itself. The number 1 is not prime. The only even prime is 2.
  • There are an infinite number of prime numbers.  Why?  Consider any finite set of primes.   Multiply this set of primes and add 1 to the product.    The resulting number is not divisible by any of the previous primes.   Since every number must have a unique prime factorization, this number is either prime or has a prime factor greater than the largest prime in the original set.
  • Every whole number has a unique prime factorization.   This is important.  That is, every whole number can be written as the product of prime numbers and this way is unique.   You probably remember making prime factory trees in grade school.  We are going to look at those now.

  • Prime Factorization (Getting to the Building Blocks of Numbers)
    To prime factorize a number (i.e. express a number as a product of primes), divide the number by factors, putting the divisor on one branch and the quotient on the other until only primes are left: Example: prime factorization of 48.

    Two Possible Prime Factorization Trees of 48

    Notice it doesn’t matter which factor we start the division with. The end result is the same.

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