Divisibility Rules - 7

Divisibility Rules - 7 exercise

• State how to use the divisibility rules of the number $7$.

  Hints

  The first step in checking if a number is divisible by $7$ is doubling the last digit of that number.

  After doubling the last digit of the number you are checking, you subtract this new number from what is left of the original number.

  For example, with $827$, you double $7$ to get $14$. You then subtract $14$ from $82$ to get $68$. Is $68$ divisible by $7$?

  Solution

  Here is how Snow White can use the divisibility rules of the number $7$:

  To check if a large number is divisible by $7$, she first takes the last digit of the number, and doubles it. In this case, with $543$ apples, she takes $3$, and doubles it to get $6$.

  Then she subtracts the new number from the rest of the original number. In this case she subtracts $6$ from $54$, and gets $48$.

  If the result is divisible by $7$, so is the original number. If it is not divisible by $7$, the original number wasn't divisible by $7$ either.

  In this case, $48$ is not divisible by $7$ because $7$ is not a factor of $48$. Therefore, $543$ is not divisible by $7$ either.

• Summarize the divisibility rule for $7$.

  Hints

  There is one mistake in each of the three steps.

  Pay careful attention to which operations should be used in each step.

  Solution

  1. The first step is to double the last digit, not triple it.
  2. In the second step, you subtract the result from the remaining digits. You do not use addition.
  3. Lastly, if the difference is divisible by $7$, then the original number is also divisible by $7$.

• Explain how to use the divisibility rule of $7$ with large numbers.

  Hints

  The divisibility rule of $7$ results in a smaller number. If that smaller number is divisible by $7$, then the original number is also divisible by $7$. If the number that the divisibility rules gives you is still very big, then you can repeat the divisibility rule over and over. Eventually you will get a number that is small enough to see if it is divisible by $7$ without having to do long division.

  It might help to write out the steps of the divisibility rule of $7$. Comparing your calculations to the steps will help you check for errors.

  Solution

  Snow White uses the divisibility rules of $7$ to check if $9827$ is divisible by $7$.

  She starts by using the divisible rules of $7$ on $9827$, just like she would with a smaller number. She takes the last digit off, doubles it, and subtracts the result from what is left of the original number.

  She takes $7$ off of $9827$. Doubling $7$, she gets $14$. She then subtracts $14$ from $982$ to get $968$.

  Snow White now has a number that is smaller than what she started with, but is still very big. She knows that she can repeat the same steps, to check if this new number is divisible by $7$. She does not need a new technique or a calculator.

  She takes $8$ off of $968$. Doubling $8$, she gets $16$. She subtracts $16$ from $96$ and gets $80$. We then have that $80$ is not divisible by $7$, so she knows that $9827$ is not divisible by $7$ either.

• Determine which offers are divisible by $7$.

  Hints

  You can use the the divisibility rules of the number $7$ to check if the offered number of each snack is divisible by $7$.

  If you get a large number after using the divisibility rules of the number $7$, you can repeat the steps to get a smaller number.

  To check if a large number is divisible by $7$, first take the last digit of the number, and double it.

  Then subtract the new number from the rest of the original number.

  If the result is divisible by $7$, so is the original number.

  Solution

  Snow White checks the offered number of each snack to see if it is divisible by $7$. She starts with the $1083$ peanuts.

  1. She takes the last digit off, doubles it, and subtracts the result from what is left of the original number. Here, Snow White takes $3$ off of $1083$. Doubling $3$, she gets $6$. She then subtracts $6$ from $108$ to get $102$.
  2. She is not sure if $102$ is divisible by $7$, so she repeats her steps. Snow White takes $2$ off of $102$, and doubles it to get $4$. She subtracts $4$ from $10$ to get $6$.
  3. Six is not divisible by $7$, so she knows that $1083$ is not divisible by $7$. So she cannot buy the peanuts for her friends.

  Similarly, she checks the other snacks.

  • $596$ is not divisible by $7$, so she cannot buy the cherry tomatoes.
  • $826$ and $7406$ are divisible by $7$, so she can buy the almonds and the blueberries.

• Explain what it means if a number is divisible by another number.

  Hints

  In order for two numbers to be called divisible, there cannot be a remainder after you divide them.

  Do the divisibility rules take you a longer time to do than doing long division?

  Division is reverse multiplication, and factors are numbers we can multiply together to get another number.

  Solution

  1. Given two numbers, if you can divide the first by the second without any remainder, the first number is divisible by the second. ✓ True.
  2. One number is divisible by another, even if you get a remainder when you do the division operation. ✗ False. If you get a remainder, the number is not divisible by the number you used.
  3. If a large number of apples is divisible by $6$, you can put all the apples into 6 even piles, and have no apples left over. ✓ True. This is equivalent to saying "if a large number is divisible by six, you will not get a remainder when dividing it by 6".
  4. A divisibility rule is a very long way of determining whether a given number is divisible by a fixed divisor without performing the division. ✗ False. A divisibility rule is a shorthand, or typically faster, way of figuring out divisibility without actually dividing.
  5. If one whole number is divisible by another number, then the second number is a factor of the first number. ✓ True. Factors are numbers we can multiply together to get another number.

  By the way, zero is divisible by every number because zero divided by any number is zero. Therefore, no remainder will ever exist.

• Determine which numbers are divisible by $7$.

  Hints

  You can use the the divisibility rules of the number $7$ to check if the offered number of each supply is divisible by $7$.

  When using the divisibility rules of the number $7$ with large numbers, you will need to perform the steps several times in a row.

  To check if a large number is divisible by $7$, first take the last digit of the number, and double it.

  Then subtract the new number from the rest of the original number.

  If the result is divisible by $7$, so is the original number.

  Solution

  The dwarfs start off by checking if $89273$ nails are divisible by $7$.

  1. They take the last digit off, double it, and subtract the result from what is left of the original number. Doubling $3$, they get $6$. Subtracting $6$ from $8927$, they get $8921$.
  2. This number is still quite large, so they repeat the process until they get a one or a two digit number. Doubling $1$, they get $2$. Subtracting $2$ from $892$, they get $890$. Doubling $0$, they get $0$. Subtracting $0$ from $89$, they get $89$.
  3. They know that $89$ is not divisible by $7$. So $89273$ must not be divisible by $7$, either.

  Using the same strategy, they find:

  ✓ The $83489$ screws are divisible by $7$.

  ✓ The $39823$ sticks are divisible by $7$.

  ✗ The $72382$ planks are not divisible by $7$.

  ✓ The $29827$ bolts are divisible by $7$.

  ✗ The $19723$ washers are not divisible by $7$.