Divisibility Rules - 7 04:21 minutes

Video Transcript

Transcript Divisibility Rules - 7

Mirror, mirror, on the wall, who's the cleverest of them all? The evil witch, Snow White, or her 7 friends Stay tuned to see how this story ends The witch has a notion she thinks is terrific to make a potion that's also a soporific. With Snow White in dreamland, the evil witch can unveil her plan. There's just one glitch, a hitch for the witch To avoid a one-way ticket to heaven, Snow White leans on the Divisibility Rules for the number 7.

In an attempt to trick Snow White, the witch offers her a basket filled with 15, delicious-looking apples. Snow White doesn’t know the apples are laced with a sleeping potion, but she rejects them regardless. Why? Because she can’t divide the 15 apples among the 7 dwarfs evenly, and she doesn't play favorites. The witch is not discouraged. So the very next day, she returns. This time, she has a cart full of apples. The witch doubts that Snow White can calculate such a large quotient quickly and will simply decide to accept the cart and its poisonous contents. The witch proudly declares that she has 543 apples, more than the dwarves and Snow White can ever eat. Again, Snow White refuses because she can’t divide the number of apples evenly into groups of 7. How did she determine this so quickly?

Divisibility by 7

Snow White is a master of the divisibility rule for the number 7, so she doesn’t have to always rely on long division. To check if a number is evenly divisible by 7: Take the last digit of the number, double it Then subtract the result from the rest of the number If the resulting number is evenly divisible by 7, so is the original number. Let’s try the trick on the number of apples in the cart, 543. The last digit is 3, double that to make 6, subtract from 6 from the remaining digits. 54 minus 6 is equal to 48. 48’s not evenly divisible by 7, so 543 isn't evenly divisible by 7 either.

Let's check, just to make sure. 7 goes into 54 seven times. Subtract 49 from 54, bring down the 3, 7 goes into 53 seven times, subtract 49 from 53, which leaves us with a remainder of 4. So we were right! 543 isn't evenly divisible by 7!

Foiled again. What's an evil witch to do? Has Snow White simply outsmarted her?

The evil witch doesn't give up. She gathers all the apples in the kingdom, 2478 to be exact, and delivers them to Snow White. Let’s see.Ok. The last digit is 8. Double it, and we get 16. Subtract 16 from 247. The difference is 231. That’s still a big number, so we just do the same steps again. Double the last digit, that's equal to 2 and 23 minus 2 is equal to 21. 21 is evenly divisible by 7, so the ginormous pile of apples must also be evenly divisible by 7! 7 goes into 24 three times, subtract 21 from 24, bring down the 7, 7 goes into 37 five times.

Subtracting from 37 gives us 2 and 7 goes into 28 exactly 4 times. Whaddya know? Snow White was correct! 2478 IS evenly divisible by 7!

While we were busy calculating, 77 pies are now ready and waiting. Prepared by Snow White with love and care, her pies are famous far, wide and everywhere. And because she's so super sweet, she offers the witch a pie that can't be beat.

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

    Tipps

    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 $7$?

    Lösung

    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.

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

    Tipps

    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.

    Lösung

    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.

  • Summarize the divisibility rule for $7$.

    Tipps

    There is one mistake in each of the three steps.

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

    Lösung
    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 $7$.
  • Determine which numbers are divisible by $7$.

    Tipps

    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.

    Lösung

    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 $7$.

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

    Tipps

    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.

    Lösung

    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 $7$.

    Tipps

    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.

    Lösung

    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.