In our previous lesson, we discussed the divisibility rules for 2, 3, 4, 5, 6, 9, and 10. In this lesson, we are going to talk about the divisibility tests for numbers 7, 11, and 12. The reason why I separated them is that the divisibility rules for 7, 11, and 12 are a little bit more advanced. However, I promise you that after learning their respective rules and applying them to some practice problems, you will realize that they are not that difficult. In fact, they are actually fun!
Divisibility Rule for 7
Regla: Take the last digit and cross it out from the original number. Then double it. Subtract it from the “new” number which is the original number excluding the last digit. If the difference is divisible by 7, then the original number must also be divisible by 7. If at first application, the result is not obviously divisible by 7, you can repeat the process as needed until you reach a two-digit number which can easily be determined if it is divisible by 7 or not.
Ejemplo 1: True or False. The number 6,895 is divisible by 7.
Solution: Let’s take the last digit of 6,89{color{red}5} which color{red}5 then double it, thus 2({color{red}5})=10. Now, subtract the “new” number (old number excluding the last digit) by twice the last digit, we have 689-10=679. Is 679 divisible by 7? We can perform long division. But the good thing is that we can perform the process again and again until we reach a two-digit number because it is much easier to know if it’s divisible or not by 7.
Let’s repeat the process one more time and see what we will get. Remember we ended up at 679 from the last step. Moving on, the last digit of 67{color{red}9} is color{red}9. If we double it, we get 2({color{red}9})=18. The remaining number formed when we get rid of the last digit is 67. If we subtract 67 by 18, we obtain 67-18=49.
Since 49 is divisible by 7, therefore the original number 6,895 must also be divisible by 7. So, the answer is True. ✔︎
Ejemplo 2: Multiple Choice. Which number is divisible by 7?
Note: There is only one correct answer.
A) 18,046
B) 11,749
C) 20,704
D) 21,011
I understand that the procedure can be tricky at first but the more you use it, the more it becomes much easier. Below are easy to follow steps that I hope can help to cement in your memory.
Steps to Check for the Divisibility of 7
- Drop the last digit of the number then double the digit that we dropped.
- Subtract it from the new number formed by removing the last digit of the original number.
- Repeat the process until the number is reduced to two digits.
- If the two-digit number is divisible by 7, then the original number is divisible by 7. Otherwise, it is not.
Solution: In an actual multiple questions test, you may want to randomly select an option (a letter) to solve because it is possible that you can stumble upon the correct answer right away, therefore, saving you a lot of time. But in this lesson, we will go from A to D for the sake of practice.
◉ Testing Option A: 18,046
Drop the last digit of 18,046 which becomes 1,804 then double the digit that we dropped, so we have 2(6)=12.
Subtract the new number by the double of the last digit: 1,804 - 12 = 1,792. We have reduced the original five-digit number to a four-digit number. Remember, we want it to be reduced to a two-digit number. Let’s repeat the process.
Drop the last digit of 1,792 which becomes 179 then double the digit the we dropped, so we have 2(2)=4.
Subtract the new number by the double of the last digit: 179 - 4 = 175. We have reduced it now to a three-digit number. Let’s do it one more time!
Drop the last digit of 175 which becomes 17 then double the digit that we removed, thus 2(5)=10.
Subtract the new number by twice the last digit: 17-10=7.
Since color{red}7 is divisible by 7, then the original number which is 18,046 is also divisible by 7. So, option A is the correct answer. ✔︎
The final answer is option A.
I will leave it to you as an exercise as to why options B, C, D are NOT divisible by 7. However, I will still provide you with a shortened solution below. I highly encourage you to perform the exercise not only for more practice but also it is as satisfying to show that a number is not divisible by 7.
You Try!
◉ Testing Option B: 11,749
Click Here to Show Solution- Original number: 11,749
- 1,174-2(9)=1,174-18=1,156
- 115-2(6)=115-12=103
- 10-2(3)=10-6=4
Since color{red}4 is not divisible by 7 then 11,749 is also not divisible by 7. ✘
◉ Testing Option C: 20,704
Click Here to Show Solution- Original number: 20,704
- 2,070-2(4)=2,070-8=2,062
- 206-2(2)=206-4=202
- 20-2(2)=20-4=16
Since color{red}16 is not divisible by 7 then 20,704 is also not divisible by 7. ✘
◉ Testing Option D: 21,011
Click Here to Show Solution- Original number: 21,011
- 2,101-2(1)=2,101-2= 2,099
- 209-2(9)=209-18=191
- 19-2(1)=19-2=17
Since color{red}17 is not divisible by 7 then the original number which is 21,011 is not divisible by 7 as well. ✘
Ejemplo 3: Select all that apply. Which numbers are divisible by 7?
Note: There can be more than one answer.
A) 5,544
B) 3,110
C) 54,810
D) 34,125
Solution: I am sure that at this point you have already mastered the steps on how to check if a number is divisible by 7 or not. With that said, I will be using a shortened solution.
◉ Testing Option A: 5,544
We are testing if 5,544 is divisible by 7.
554-2(4)=554-8=546
54-2(6)=54-12=42
Because 42 can be divided by 7 then the original number 5,544 is also divisible by 7. ✔︎
◉ Testing Option B: 3,110
We are checking if 3,110 is divisible by 7.
311-2(0)=311-0=311
31-2(1)=31-2=29
Since 29 cannot be divided by 7 then the original number 3,110 is not divisible by 7 either. ✘
◉ Testing Option C: 54,810
Let’s examine if 54,810 is divisible by 7.
5,481-2(0)=5,481-0=5,481
548-2(1)=548-2=546
54-2(6)=54-12=42
The algorithm has reduced the original number into a two-digit number which is 42 that is divisible by 7. It means that the original number 54,810 must also be divisible by 7. ✔︎
◉ Testing Option D: 34,125
Let’s determine if 34,125 is divisible by 7.
3,412-2(5)=3,412-10=3,402
340-2(2)=340-4=336
33-2(6)=33-12=21
We have reduced the original five-digit number into a two-digit number 21 that is divisible by 7. It implies that the original number 34,125 should be divisible by 7 as well. ✔︎
So in summary, options A, C, D son divisibles por 7.
Divisibility Rule for 11
Regla: From the left to right of a number, take the first digit, and attach an addition symbol to its left. Then subtract it by the next digit, then add the result by the third digit, and subtract again the result by the fourth digit, and so on and so forth. If the answer is divisible by 11, then the original number is divisible by 11.
Condensed Rule: Alternately add and subtract the digits of a number from left to right. If the answer is divisible by 11, then the original number is divisible by 11.
Standard Rule: Take the alternating sum of the digits of a number. If the result is a multiple of 11, the number is divisible by 11.
NOTA: All the rules above mean the same thing. The first two rules are more instructive in nature while the last one is the rule that you may encounter in your textbook or being taught by your teacher.
Ejemplo 1: True or False. The number 9,581 is divisible by 11.
The rule is actually quite simple. We will add and subtract, then repeat the pattern until all the digits of the number are assigned with a plus and minus symbols from left to right. After setting it up, we simplify it. If the result is a multiple of 11, then the original number is also divisible by 11.
Here is the set-up:
+ 9-5 + 8-1
Step 1: +9-5=4
4 8 +-1
Step 2: 4+8=12
12-1
Step 3: 12-1=11
11
Since the final result is 11 and a multiple of 11, then the original number which is 9,581 is divisible of 11. Thus, our final answer is True. ✔︎
Ejemplo 2: Multiple Choice. Which number is divisible by 11?
Note: There is only one correct answer.
A) 98,517
B) 79,829
C) 82,709
D) 50,453
We will check the divisibility of each number from option A to option D.
◉ Checking Option A: 98,517
Let’s set it up by taking the alternating sum of the digits of the number.
9-8+5-1+7
Then, we simplify.
(9-8)+5-1+7
1 + 5-1 + 7
(1+5)-1+7
6 1 7-+
(6-1)+7
5 + 7
12
The final result is 12 which is not a multiple of 11. Therefore, the original number 98,517 is not divisible by 11. ✘
◉ Checking Option B: 79,829
Set it up by writing the alternating sum of the digits.
7+9-8+2-9
Simplificar.
(7+9)-8+2-9
16-8 + 2-9
(16-8)+2-9
8 2 +-9
(8 + 2) -9
10-9
1
Since the final answer large{(1)} is not divisible by 11, therefore the original number 79,829 is also not divisible by 11. ✘
◉ Checking Option C: 82,709
We first construct the alternating sum of the digits of the number.
8-2+7-0+9
Then simplify from left to right. No need to worry about the Order of Operations since we are only dealing with addition and subtraction.
(8-2)+7-0+9
6 + 7-0 + 9
(6+7)-0+9
13 0 9-+
(13-0)+9
13 + 9
12
Since the final result is 12 which is a multiple of 11, it implies that the original number 82,709 is divisible by 11. Therefore, the final answer is C. ✔︎
☞ There is no need to check for Option D because we have already found the correct answer.
The final answer is option C.
Ejemplo 3: Which numbers are divisible by 11? Select all that apply.
Note: There can be more than one answer.
A) 69,245
B) 73,186
C) 843,210
D) 918,071
Solución:
◉ Testing Option A: 69,245 if it is divisible by 11
6-9+2-4+5
{color{red}6-9}+2-4+5
-3+2-4+5
{color{red}-3+2}-4+5
-1-4+5
{color{red}-1-4}+5
-5 + 5
0
Since 0 is a multiple of 11, therefore 69,245 is divisible by 11. ✔︎
◉ Testing Option B: 73,186 if it is divisible by 11
7-3+1-8+6
{color{red}7-3}+1-8+6
4 + 1-8 + 6
{color{red}4+1}-8+6
5 8 6-+
{color{red}5-8}+6
-3 + 6
3
Since 3 is not a multiple of 11, thus 73,186 is not divisible by 11. ✘
◉ Testing Option C: 843,210 if it is divisible by 11
8-4+3-2+1-0
{color{red}8-4}+3-2+1-0
4+3-2+1-0
{color{red}4+3}-2+1-0
7-2 + 1-0
{color{red}7-2}+1-0
5 1 +-0
{color{red}5+1}-0
6-0
6
Since 6 is not a multiple of 11, hence 843,210 is not divisible by 11. ✘
◉ Testing Option D: 918,071 if it is divisible by 11
9-1+8-0+7-1
{color{red}9-1}+8-0+7-1
8+8-0+7-1
{color{red}8+8}-0+7-1
16-0 + 7-1
{color{red}16-0}+7-1
16 7 +-1
{color{red}16+7}-1
23-1
22
Since 22 is a multiple of 11, it implies that 918,071 is divisible by 11. ✔︎
In summary, options A y D son divisibles por 11.
Divisibility Rule for 12
Regla: A number is divisible by 12 if it is both divisible by 3 and 4.
- Un número es divisible por 3 if the sum of its digits is divisible by 3.
- Un número es divisible por 4 if the last two digits of the number are divisible by 4.
Ejemplo 1: True or False. The number 7,512 is divisible by 12.
Solución:
The first step is to check if it is divisible by 3. We will first add all the digits of the number of 7,512.
7,512
7+5+1+2=15
Since 15 is divisible by 3, therefore 7,512 is also divisible by 3.
The last step is to test if the number formed by the last two digits of the original number is divisible by 4, then it is divisible by 4.
7,5{color{red}12}
Since 12 is divisible by 4, then 7,512 is divisible by 4.
Therefore, because the original number 7,512 is both divisible by 3 and 4, then it must be divisible by 12. ✔︎
Ejemplo 2: Multiple Choice. Which number is divisible by 12?
Note: There is only one correct answer.
A) 527,037
B) 981,128
C) 746,936
D) 49,9920
Solución:
There is a faster way to test for the divisibility of 12. Remember, a number is divisible by 12 if 3 and 4 can both divide it. Since it is much quicker to test for the divisibility of 4 que 3 because for the former you just have to look at the last two digits of the number and check if it is a multiple of 4, and the latter will take slightly more time because you will have to add all the digits of the number and check if the sum is divisible by 3. Therefore, we will check first for the divisibility of 4 followed by the divisibility of 3. The other way around is a little bit more time-consuming.
◉ Testing Option A: 527,037 for divisibility of 12
The last two digits of 527,037 is color{red}37 is not a multiple of 4. Therefore, it is not divisible by 4. There is no need to check for the divisibility of 3 since it fails on one of the two requirements. Thus, 527,037 is not divisible by 12. ✘
◉ Testing Option B: 981,128 for divisibility of 12
The last two digits of 981,128 is color{red}28 which is a multiple of 4 that makes it divisible of 4. Now let’s check if it is divisible by 3 by adding all its digits, thus 9+8+1+1+2+8=29. Since, the sum 29 is not divisible by 3, then the number itself is also not divisible by 3. Because 981,128 cannot be divided by en 3 and 4, that means the two requirements are not met, hence the original number is not divisible by 12. ✘
◉ Testing Option C: 746,936 for divisibility of 12
The number color{red}36 is the last two digits of 746,936. And it is a multiple of 4 which makes the original number divisible by 4. Now for divisibility of 3. Add all the digits of 746,936, we get 7+4+6+9+3+6=35. The sum of the digits is not divisible by 3. It follows that the number is also not divisible by 3. Because one of the two required conditions are not met (both are not true), then 746,936 is not divisible by 12. ✘
◉ Testing Option D: 49,9920 for divisibility of 12
The number 20 is the last two digits of 49,9920 which is clearly a multiple of 4, thus makes 49,9920 divisible by 4. Adding up all the digits of the number: 4+9+9+9+2+0=33. The sum 33 can be divided by 3 and so 49,9920 is divisible by 3. Since, the original number is both divisible by 3 and 4, it must also be divisible by 12. ✔︎
The final answer is option D.
Ejemplo 3: Which numbers are divisible by 12? Select all that apply.
Note: There can be more than one answer.
A) 344,888
The number color{red}88 is the last two digits of 344,888 which is clearly a multiple of 4, thus divisible by 4.
The sum of the digits of 344,888 is calculated as 3+4+4+8+8+8=35. But 35 is obviously not divisible by 3.
Since 344,888 is only found to be divisible by 4 but not by 3, failing one of the two requirements implies that the original number is not divisible by 12. ✘
B) 521,340
The last two digits of 521,340 formed the number color{red}40 which is a multiple of 4, thus can be divided by 4.
Adding up its digits we get 5+2+1+3+4+0=15. The sum 15 is divisible by 3.
Since 521,340 is both divisible by 3 and 4, then it must be divisible by 12. ✔︎
C) 842,652
The number color{red}52 is the last two digits of the number which is clearly divisible by 4.
The sum of the digits is 8+4+2+6+5+2=27. The number 27 is divisible by 3.
Since 842,652 are both divisible by 3 and 4, then it should also be divisible by 12. ✔︎
D) 676,968
The last two digits color{red}68 are divisible by 4.
The sum of the digits 6+7+6+9+6+8=42 is divisible by 3.
Because the original number can both be divided by 3 and 4, then it must also be divisible by 12.
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