5.4. Divisibility

  • A Digit is a symbol used to represent a number and is used in combination with other digits to create other numbers.

    • The digits we are most familiar with are the Arabic Numerals: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. These are the "number" keys on our calculator.

  • The Digit Sum of a number is found by adding together the individual digits used to create the number. For example, the digit sum of 351 is 3 + 5 + 1 = 9. Finding the sum of digits will be handy to know later in this lesson.

  • A number is said to be Divisible by another number if it can divide into it exactly (there is no remainder). Numbers are divisible by all of their factors.

  • An Even Number is any whole number that is divisible by 2. Zero is considered an even number. Numbers ending in 0, 2, 4, 6, and 8 are even.

  • An Odd Number is any whole number that is NOT divisible by 2. When divided by 2, the remainder would be 1. Numbers ending in 1, 3, 5, 7, and 9 are odd, but not as odd as this tutor.


DIVISIBILITY RULES

Divisible By: The Rule: An Example:
2 It is divisible by 2 if the number is even (last digit is 0, 2, 4, 6, 8). 31,528 is divisible by 2 since it
ends with the even digit 8.
3 It is divisible by 3 if, when the digit sum is divisible by 3, then the original number is. Repeat with the digit sum itself, if necessary. 70,221 is divisible by 3 since: 7 + 0 + 2 + 2 + 1 = 12 and 12 is divisible by 3 since: 1 + 2 = 3, which is also divisible by 3.
5 It is divisible by 5 if the last digit is either a 0 or a 5. 6,825 is divisible by 5 since it ends with 5.
6 It is divisible by 6 if it is divisible by 2 (even) and by 3. 2,334 is divisible by 6 since it is even and the digit sum (2 + 3 + 3 + 4 = 12) is divisible by 3.
9 It is divisible by 9 if, when the digit sum is divisible by 9, then the original number is. Repeat with the digit sum itself, if necessary. 52,416 is divisible by 9 since 5 + 2 + 4 + 1 + 6 = 18 and 18 is divisible by 9 (1 + 8 = 9).
10 It is divisible by 10 if the last digit is 0. 490 is divisible by 10 since it ends with a 0.

There are other rules of divisibility for other numbers, but I find they are either not too useful, or just too complicated to use and/or remember.

With these rules, finding the prime factorization of a number is easier.

Basic Strategy:

  1. Try thinking of numbers that can split the number into others that are close in size to each other. For example, with 16, I would not try 2 and 8 first, but rather 4 and 4. With 50, I would not try 2 and 25, but I'd try 5 and 10 first. It really does not matter, in the long run, if you do this. I can start by dividing by 2. You would get longer factor trees this way, and we want to minimize our work, if possible.

  2. Now, if presented with a large number, mentally factoring it may not be easy, so use the divisibility rules to try to factor out a "large" number like 9, 10 (if it ends with 0) or 6 first. Then try 2 (if itís even) and 5 (if it ends with a 5). When you tried 9, and it did not work, but got something that is divisible by 3, then you can use that information to factor the number.

  3. Keep factoring the "branches" (or original number). You may need to now try other primes as guesses, like dividing the number by 11, by 13, by 17, etc. Continue this until you get only prime numbers. Circle the primes (the "leaves").

  4. Displaying these primes as a product gives you the prime factorization.

In step 3, a question may come up: when do you stop trying to guess what prime numbers would divide into the number?

The Prime Factor Test: To find candidates to use to find the prime factors of a number, you only need to try those that are less than or equal to the square root of the number.

Example 9: What is the largest number you should use to try and factor 221? What is its prime factorization?

SOLUTION: Our largest candidate for guessing what factor would work is:

So the biggest number you need to try to figure out what numbers divide into 221 is 14. The largest prime would be 13.

Using our divisibility rules:

10 does not work since 221 does not end with a 0.
9 (and 3) will not work since the digit sum is: 2 + 2 + 1 = 5.
5 does not work since 221 does not end with a 5.
2 will not work. 221 is not even.
Since 2 and 3 did not work, 6 will not work.

Now, start dividing 221 by some primes (use a calculator, it is faster):

221 ÷ 7 ≈ 31.57…<not divisible by 7>

221 ÷ 11 ≈ 20.909…<not divisible by 11>

221 ÷ 13 = 31.57…&ltIt works!>

Using a factor tree:

Since 13 and 17 are both prime, we are done.

The prime factorization of 221 is 13 x 17.

Example 10: Find the prime factorization of 1176.

SOLUTION: The digit sum is: 1 + 1 + 7 + 6 = 15 which is divisible by 3, and it is even, so we can divide it by 6 <use a calculator>

1176 ÷ 6 = 196.

196 is even, so divide by 2 :

198 ÷ 2 = 98.

This is even, so divide by 2 again: 98 ÷ 2 = 49.

This should be enough information to create our factor tree:


The prime factorization of 1176 is 2 x 2 x 2 x 3 x 7 x 7 or 22 x 3 x 72

Note: This is NOT the only way of creating the factor tree.