Divisibility Rules - 4, 5, 8, 10

Divisibility Rules - 4, 5, 8, 10 exercise

• Decide which number is divisible by $4$.

  Hints

  Let's have a look at an example: $1232$

  First we divide this number by $2$ to get $616$. Once more we divide it by $2$. The result is $308$.

  Let's check the result: $4\times 308=1232$ $~~~~~$✓

  $32$ is divisible by $4$ because $32\div4=8$.

  Only even numbers are divisible by $2$.

  Solution

  Because the multiples of $4$ are:

  $4$, $8$, ... , $104$, $108$, $112$, $116$, $120$, ...

  We can use the Divisibility Rule for $4$:

  • A number is divisible by $4$ if the last two digits are divisible by $4$.

  • Or, if the number can be divided by $2$ two times and the quotient is a whole number.

  Let's have a look at Billy Bonka's problem.

  • $1516$ blueberry balls: The last two digits are $16$ and sure this is divisible by $4$. So $1516$ is divisible by $4$, too.
  • $1035$ caramel cubes: Because the last digit is odd, this number can't be divided by $2$ and thus can't be divided by $4$.
  • $1600$ strawberry strips: $00$ is divisible by $4$ because $0/4 = 0$ and thus $1600$ is divisible by $4$.

• Find the correct statements for divisibility rules for 4, 5, 8, and 10.

  Hints

  For example: $35$ is divisible by $5$.

  On the other hand $40$ is divisible by $5$, as well as by $10$.

  $12$ is divisible by $4$, but you can't divide it by $8$.

  • $20\div 2=6$
  • $10\div 2=3$

  The second time we divided $20$ by $2$ resulted in a whole number, but $20$ is not divisible by $8$.

  Solution

  Divisibility by $10$:

  • Each number ending with $0$ is divisible by $10$.

  Divisibility by $5$:

  • Each number ending with $5$ or $0$ is divisible by $5$.

  Divisibility by $4$

  • A number is divisible by $4$ if the last two digits are evenly divisible by $4$.
  • ... or ... if the number is divisible by $2$ twice and the quotient is a whole number.

  Divisibility by $8$

  • A number is divisible by $8$ if the last three digits are evenly divisible by $8$.
  • ... or ... if the number is divisible by $2$ three times and the quotient is a whole number.

• Explain why the numbers are divisible using the rules for divisibility.

  Hints

  Keep the rules beside in mind.

  $110$ is divisible by $5$ and $748$ is divisible by $8$.

  Solution

  Let's practise those divisibility rules:

  $110$ is divisible by $5$ and by $10$: ... by $10$ because it ends with $0$.

  $748$ is divisible by $4$ and by $8$: ... by $4$ because the last two digits are evenly divisible by $4$. $48\div 4=12$.

  $1656$ is divisible by $4$ and by $8$ as well: ... by $8$ because the last three digits are evenly divisible by $8$: $656\div 8=82$.

  $1965$ is only divisible by $5$ because it ends with $5$.

  Keep the following in mind:

  • Each number divisible by $8$ is also divisible by $4$.
  • Each number divisible by $10$ is also divisible by $5$.

• Determine if the numbers are divisible by $4$ or $5$.

  Hints

  A number is divisible by $4$ if the last two digits are divisible by $4$.

  Or ... if the number is divisible by $2$ twice and the quotient is a whole number.

  For example $116$ is divisible by $4$ but $114$ isn't.

  Each number ending with $5$ or $0$ is divisible by $5$.

  A number divisible by $4$ is at least even.

  Solution

  Divisibility by $4$ can be determined as follows:

  • The last two digits are divisible by $4$.
  • The number is divisible by $2$ twice and the quotient is a whole number.
  • Only even numbers are divisible by $4$ but not all even numbers are divisible by $4$.

  So $316$, $1224$, and $676$ are divisible by $4$.

  Let's have a look at $670$: Is this number also divisible by $4$? Let's check it by dividing by $2$ twice: $670\div 2=335$. The result, $335$, is odd and thus not evenly divisible by $2$. Therefore, $670$ isn't divisible by $4$.

  Divisibility by $5$ can be determined as follows:

  • The number ends with $5$ or $0$.

  So we can consider that $670$, $315$, and $1230$ are divisible by $5$.

• Calculate in which cases numbers are divisible by $2$.

  Hints

  If a number is divided by $2$ and doesn't have a remainder, it also means it's evenly divisible or divisible by $2$.

  Write down the multiples of $2$.

  The last digit of any number divisible by $2$ is $0$, $2$, $4$, $6$ or $8$.

  Solution

  The multiples of $2$ are:

  $2$, $4$, $6$, $8$, ..., $12$, $14$, $16$, ..., $102$, $104$, ...

  • Each of those numbers is even.
  • Or, in other words the last digit is divisible by $2$.
  • Or, each number ends with $0$, $2$, $4$, $6$ or $8$.

• Find the numbers that are divisible by $8$.

  Hints

  A number is divisible by $8$ if the last three numbers are divisible by $8$.

  ... or ... if the number is divisible by $2$ three times and the quotient is a whole number.

  Just look at the following:

  $340$ is divisible by $4$ because $40$ is divisible by $4$ but it isn't divisible by $8$.

  • $340\div 2=170$
  • $170\div 2=85$
  • The result is odd and thus not divisible by $2$.

  A number divisible by $8$ is divisible is at the very least divisible by $4$.

  Solution

  A number is divisible by $8$ if the last three numbers are divisible by $8$.

  Or if the number is divisible by $2$ three times and the quotient is a whole number.

  • $1352$ is divisible by $8$ because $352\div 8=44$

  • $1532$ isn't divisible by $8$, because $532=66.5$ and thus not a whole number.

  • $2224\div 2=1112 \implies 1112\div 2=556 \implies 556\div 2=278$ this is a whole number. Thus $2224$ is divisible by $8$.

  • $2220\div 2=1110 \implies 1110\div 2=555$. $555$ is an odd number and thus not evenly divisible by $2$. So we can't divide $2220$ by $2$ three times and get a whole number. $2220$ isn't divisible by $8$ (but is divisible by $4$).

  • If you know that a three digit number is divisible by $8$ you can put as many digits in front as you want. So $72224$ is also divisible by $8$ because $2224$ is divisible by $8$.

  • $2202$ can't be divisible by $8$ because $2202\div 2=1101$ results in an odd number.

  • $22274\div 2=11137$ is also an odd number. So also isn't divisible by $8$.