Divisibility Rules - 3, 6, 9

Divisibility Rules - 3, 6, 9 exercise

• Recall the divisibility rules for 3, 6, and 9.

  Hints

  The first step in the divisibility rules for $3$ and $9$ is to sum all of the digits in the number.

  A number that is divisible by $2$ and $3$ is also divisible by $6$.

  Solution

  Divisibility Rule for 3

  The first step in checking if a number is divisible by three is to sum all the digits. If the resulting sum is divisible by $3$, then the original number is also divisible by $3$. Here's an example with the number $348$:

  $3 + 4 + 8 = 15$.

  $15$ is divisible by $3$, so $348$ must also be divisible by $3$.

  $~$

  Divisibility Rule for 6

  If a number is divisible by $2$ and $3$, then it is also divisible by $6$. To check if a number is divisible by two, look at its last digit. If the last digit is $0$, $2$, $4$, $6$ or $8$, then the number is divisible by $2$. To check if a number is divisible by three, follow the instructions above.

  For example, the number $348$ is divisible by $2$ because the last digit is $8$. We already found that it is also divisible by $3$, so it must be divisible by $6$ as well.

  $~$

  Divisibility Rule for 9

  Just like the divisibility rule for $3$, the first step in checking if a number is divisible by $9$ is to sum all the digits. If the resulting sum is divisible by $9$, then the original number is divisible by $9$. Using the number $348$ as an example:

  $3 + 4 + 8 = 15$.

  $15$ is not divisible by $9$, so neither is $348$.

• Decide if numbers are divisible by 2, 3, 6, or 9.

  Hints

  Try using the divisibility rules for $3$, $6$, and $9$ to check if a number is divisible by those numbers.

  $9$ and $6$ are both divisible by $3$.

  Therefore, if a number is not divisible by $3$, then it cannot be divisible by $6$ or $9$.

  It'll be fastest to start by checking if each number is divisible by $3$. If it isn't divisible by $3$, then you already know that it's not divisible by $3$, $6$, and $9$.

  Solution

  Divisibility Rule for 3

  The first step in checking if a number is divisible by three is to sum all the digits. If the resulting sum is divisible by $3$, then the original number is also divisible by $3$. For example, with the number $128$:

  $1 + 2 + 8 = 11$

  $11$ is not divisible by $3$, so $128$ is not divisible by $3$ either.

  $~$

  Divisibility Rule for 6

  If a number is divisible by $2$ and $3$, then it is also divisible by $6$. To check if a number is divisible by two, look at its last digit. If the last digit is $0$, $2$, $4$, $6$, or $8$, then the number is divisible by $2$. $128$ is divisible by $2$ because the last digit is $8$. But we already found that is is not divisible by $3$. So it cannot be divisible by $6$ either.

  $~$

  Divisibility Rule for 9

  Just like the divisibility rule for $3$, the first step in checking if a number is divisible by $9$ is to sum all the digits. If the resulting sum is divisible by $9$, then the original number is divisible by $9$. So using the number $128$ as an example:

  $1 + 2 + 8 = 11$

  $11$ is not divisible by $9$, so neither is $348$.

• Explain how Olivia can level up if she has 1356 points.

  Hints

  Look at this example, using the number $1234$ and the sum of its digits:

  $\quad~~~1+2+3+4=10$

  This sum isn't divisible by $3$ and thus $1234$ is neither divisible by $3$ nor by $6$ nor by $9$.

  $1356$ is divisible by $2$ of the following numbers: $2$, $3$, $6$, and $9$.

  Solution

  For checking divisibility, follow these steps:

  1. Check the divisibility rule for $3$: The sum of the digits $1+3+5+6=15$ must be divisible by $3$. It sure is: $15\div 3=5$. Thus $1356$ is divisible by $3$ too.
  2. Is the given number even or odd? $1356$ ends with $6$, an even number, so $1356$ is also even and thus divisible by $2$. If any number is divisible by $2$ as well as $3$, it's divisible by $6$ too.
  3. Again take a look at the sum of the digits: $15$. It's not evenly divisible by $9$. Therefore, we can conclude that $1356$ is not divisible by $9$.

  That means Olivia could either improve her Creature's health or attack level.

• Calculate the different ways in which Olivia’s grandmother can level up if she has 5634 points.

  Hints

  The digit sum is the sum of all individual digits of a given number. Let's have a look at an example, $2345$:

  $\quad~~~2+3+4+5=14$

  A number is divisible by $3$ (or $9$) if the digit sum is divisible by $3$ (or $9$).

  Even numbers end in $0$, $2$, $4$, $6$, or $8$.

  All even numbers are divisible by $2$.

  Solution

  You want to check the divisibility of the number $5643$.

  1. First, sum the digits: $5+6+4+3=18$. This sum is divisible by $3$ as well as $9$. Thus you can conclude that the number $5643$ is divisible by $3$ and $9$.
  2. Next, take a look at the last digit: It's $6$ and thus even. $5643$ is divisible by $2$.
  3. Together with the divisibility by $3$ you know that $5643$ is divisible by $6$, too.

  Great job by Olivia's Grandma. She could improve all of her Creature's levels: health, attack and defense. She just has to decide which one!

• Determine if $348$ is divisible by 3, 6, or 9.

  Hints

  You can use the divisibility rules for $3$, $6$, and $9$ to complete this task.

  The divisibility rules for $3$ and $9$ are similar.

  The first step in the divisibility rules for $3$ and $9$ is to sum all of the digits in the number.

  Each number which is divisible by $2$ and $3$ is divisible by $6$, too.

  Solution

  Divisibility Rule for 3

  The first step in checking if a number is divisible by three is to sum all the digits. If the resulting sum is divisible by $3$, then the original number is also divisible by $3$. For example, with the number $348$:

  $3 + 4 + 8 = 15$

  $15$ is divisible by $3$, so $348$ must also be divisible by $3$.

  $~$

  Divisibility Rule for 6

  If a number is divisible by $2$ and $3$, then it is also divisible by $6$. To check if a number is divisible by two, look at its last digit. If the last digit is $0$, $2$, $4$, $6$, or $8$, then the number is divisible by $2$. To check if a number is divisible by three, follow the instructions above.

  So, the number $348$ is divisible by $2$ because the last digit is $8$. We already found that is is also divisible by $3$, so it must be divisible by $6$ as well.

  $~$

  Divisibility Rule for 9

  Just like the divisibility rule for $3$, the first step in checking if a number is divisible by $9$ is to sum all the digits. If the resulting sum is divisible by $9$, then the original number is divisible by $9$. Using the number $348$ as an example:

  $3 + 4 + 8 = 15$

  $15$ is not divisible by $9$, so neither is $348$.

• Figure out the different ways Olivia can level up if she has a certain number of points.

  Hints

  Each number divisible by $6$ must be an even number.

  A number is divisible by $3$ if the digit sum is divisible by $3$.

  The rule for divisibility by $9$ is quite similar.

  You can assign three numbers to each central element.

  Solution

  Let's start with divisibility by $9$: for a number to be divisible by 9, the digit sum must also be divisible by $9$.

  • $657$ leads to the digit sum $6+5+7=18$, which is divisible by $9$. Because $657$ is odd, it's surely not divisible by $6$.
  • $234567$ leads to the digit sum $2+3+4+5+6+7=27$. And $27\div 9=3$ so $234567$ is divisible by $9$. This number is also odd and thus not divisible by $6$.
  • $3339$ gives us the digit sum $3+3+3+9=18$. Because this is divisible by $9$ we can conclude, together with the ending $9$, $3339$ is divisible by $9$ but not by $6$.

  There are still six numbers left. We determine the digit sum for each:

  • $636$ $\rightarrow$ $6+3+6=15$ which is divisible by $3$. The number is even and thus divisible by $6$.
  • $123456$ $\rightarrow$ $1+2+3+4+5+6=21$. This is divisible by $3$ too. Again, the number is even, and so we can conclude the divisibility by $6$ once again.
  • $3336$ $\rightarrow$ $3+3+3+6=15$ is divisible by $3$. Together with the ending $6$, which indicates that the number is even, this shows us that it is divisible by $6$.
  • $746$ $\rightarrow$ $7+4+6=17$. This digit sum isn't divisible by $3$, and thus $746$ is neither divisible by $6$ nor by $9$.
  • $234566$ $\rightarrow$ $2+3+4+5+6+6=26$. Using the same process as above, this number is neither divisible by $6$ nor by $9$.
  • $3335$ $\rightarrow$ $3+3+3+5=14$. This number also isn't divisible by $6$ or $9$.