Do Not Use a Calculator: Divisibility By 7

A number is divisible by 7 if the double of the unit digit, subtracted from the number without the unit digit, results in a number divisible by 7. If the number obtained is still great, repeat the process until you can check the division by 7 .

$87985\Rightarrow 8798-5\cdot 2=8788$

$8788\Rightarrow 878-8\cdot 2=862$

$862\Rightarrow 86\Rightarrow 86-2\cdot 2=82$

$82$ is not divisible by 7. So $87985$ is not divisible by 7 too.

Subtract large multiples of $7$ from $87985$ until the division is easier.

For example:

$77777$ is clearly divisible by $7$ .

$87985-77777=10208$

$7777$ is clearly divisible by $7$ .

$10208-7777=2431$

$777$ is clearly divisible by $7$ .

$2431-777=1854$

By mental arithmetic, we can see that $1854 \not\equiv 0 \pmod{7}$ , so the original $87985$ is not divisible by $7$ . Although just regular division probably would have been faster.

Mental long division:

87(000) / 7(000) = 12, remainder 3(000) so left with 3985.

Follow with 39(00) / 7(00) = 5, remainder 4(00), left with 485.

Follow: 48(0) / 7(0) = 6, remainder 6(0),

Finally: 65. 65 / 7 = 9, remainder 2. So not divisible exactly by 7.

I started with 87/7 (or 87,000/7000) leaves you with a remainder of 3 (or 3,000)----so 3,985 is left. 39/7 leaves you with 485. 48/7 leaves you with 65, and 7 cannot factor equally into 65.

In fact dividing the number directly is more time efficient in this scenario.

I have gone though different solutions of my peers and their approaches to the problem! I personally feel some methods are better than mine. But at least i want to share my knowledge! i came across this rule in a book!

A integer n is divisible by 7, 11 or 13 if and only if the difference of the number of its thousands and the remainder of its division by 1000 is divisible by 7,11 or 13

now the solution becomes: 985 - 87 = 898 By seeing the number we can come to a conclusion that the number is not divisible by 7. Moreover if in case we were asked to find the remainder also, no additional effort is required to find it. It will be the same when 898 is divided by 7 which is 2!

Split it 8 7 9 8 5. Starting from the MSB ,find remainder of each number while dividing with 7 and append the remainder with the next number with and do the same process ,continue till end. Now rem(rem(rem(rem(rem(8/7)7)9)8)5) = 5 ; not equal to zero. Hence it's not divisible by 7. Used the concept of normal division :p.I felt this as faster for any divisibility check if the divisor is a single digit.

I just clicked no

As 1001 is a multiple of 7 ( , 11 and 13), 87870 is also a multiple of 7. It left 115 which is not a multiple of 7, means that 87985 is not divisible by 7.

The divisibility rule for 7 is to double the last digit of the number then subtract it from the total from the other digits. In this case the answer would be 22. Since 22 is not divisible by 7, neither is 87985.

85 can't divisible by 7. so the correct answer is no.

Here Is the new formula you can get 100% result by using it:

Number: 87985

step 1: take first two digits(87) divide it by 7 => remainder is:3

step 2: now put 3 in front of rest of the number which is(985) making it =>3985

step 3: Repeat (step 1) until you left with last two digits which in this will be =>65

step 4: check 65/7 => not divisible yielding remainder 2 so the number is not divisible by 7

By doing these above steps you can easily get to the result without using a pen and paper just by doing it Orally!!!!!

I used a calculator

Let a(0),a(1),a(2)...be the unit's, ten's, hundred's etc. digits of a number. Then, if 3(a(1)-a(4)+a(7)-a(10)+a(13)-a(16)...)+2(a(2)-a(5)+a(8)-a(11)+a(14)-a(17)+...)+(a(0)-a(3)+a(6)-a(9)+a(12)-a(15)+...) be divisible by 7, then the number is also divisible by 7

The key is to break this integer down as the sum of smaller integers, then use modular arithmetic. Divisibility by 7 is equivalent to congruence to 0 mod 7. To start, we have:

$87985 = (70000 + 10000) + (7000) + (700 + 200) + (70 + 10) + 5$ .

Now use the fact that we can remove any integer multiple of 7 from the right-hand side, and we'll get a number that is congruent to it mod 7:

$87985 \equiv (10000) + (200) + (10) + (5) (mod 7) = 100215 (mod 7)$ .

You could also view this process as reducing the digits of 87985 to their residues mod 7. Now just repeat this until we get something managable:

$100215 = 10005 + 210 \equiv 10005 (mod 7)$ ,

$10005 \equiv 3005 (mod 7)$ , because $10 \equiv 3 (mod 7)$ ;

$3005 \equiv 205 (mod 7)$ , because $30 \equiv 2 (mod 7)$ ;

$205 \equiv 65 (mod 7)$ , because $20 \equiv 6 (mod 7)$ ,

and because $65 = 70 - 5 \equiv - 5 (mod 7) \equiv 2 (mod 7)$ , this number is not divisible by 7.

Mental long division, but don't really need the quotient, just the remainder.

87985 / 7

8 (...) / 7 remainder 1, then bring down the next digit

1 7 / 7 remainder 3

3 9 / 7 remainder 4

4 8 / 7 remainder -1 or 6

6 5 / 7 remainder 2

87985 - 7000 = 80985. 80985 - 980( this is 140 * 7) = 80005. 80005 - 70000 = 10005. 10005 - 10010 (this is 1430 * 7) = -5. -5 is not divisible by 7.

subtract 77770 = 10215 subtract 7000 = 3215 subtract 2800 = 415 subtract 350 = 65 so remainder is 2

87985

• you have 84000 ⫶ 7 => 3985
• you have 3500 ⫶ 7 => 485
• you have 490 ⫶ 7 => (5)
• 5 do not ⫶ 7

So 87985 do not ⫶ 7

The divisibility rule of 7 is as follows:

If you double the last digit and subtract it from the rest of the number and the answer is:

• 0, or
• divisible by 7

(Note: you can apply this rule to that answer again if you want)

Applying the rule here, we have:

$87895 \to 87895 - 10 = 87885 \to 87875 \to 87865 ... \to 5$ , which is indivisible by $7$ , therefore, the answer is $\boxed{\text{No}}$ .

I see a lot of long winded ways to find a solution that is easily and quickly done mentally using long division

87985/7=12569 + 2/7

Remove the last digit, double it, subtract it from the truncated original number and continue doing this until only one digit remains. If this is 0 or 7, then the original number is divisible by 7. Now checking 87985 :-

8798 - 2x5 = 8788 878 - 2x8 = 862 86 - 4 = 82 8 - 4 = 4 4 isn't divisible by 7, so 87985 is not divisible by 7

check the last digit of the number. here it is 5. the number here can be splited as 87950 + 35. Now you have to just check for 87950 which indirectly is checking 8795. now last digit of it is 5. so it can be split as 8760 + 35............ the procedure is as follows:

87985 = 87950 + 35 = 8795*10 + 35 // Now, we have to just check for 8795

8795 = 8760 + 35

876 = 820 + 56

82 = 77 + 5

=> Not a multiple f 7

87985-70000 = 17985 17985-14000 = 3985 3985- 3500 = 485 and 490 is divisible by 7 so 87985 is not divisible by 7

I found it easier. Divide 85/7. If it is exactly, we go on with the other numbers. If the last two digits fails to return exact result, then the answer is NO

8 -> 1 17 -> 3 39 -> 4 48 -> 6 65 -> 2

thus 87985 gives a remainder of 2 when divided by 7 -> No Best part? you can do all those calculation in your head

70000 is divisible by 7, so it´s 14000, then 84000 (70000 + 14000) is too, 3500 is divisible by 7, and 87500 (84000 + 3500) is too, 490 is divisible by 7, so it´s 87990 (87500 + 490), but 87990 - 7 = 87983, and if 87983 is divisible by 7, 87985 is not.

Took me about 20 seconds to work out, not sure how everybody else took so long.

If you simply use long division (a method), you can quickly find the answer, you don't even need a pen and paper.

7 / 87985

You first divide by the first digit. 8 / 7 leaves a remainder of 1. (8 goes into 7 once leaving 1 left over, also known as the remainder)

You then place the previous remainder in front of the next digit. Next digit is 7, remainder is 1, therefore next value is 17. 17 / 7 leaves a remainder of 3. (7 goes into 17 twice, leaving 3 left over)

Repeat until you get to the end.

39 / 7, remainder = 4 48 / 7, remainder = 6 65 / 7, remainder = 2

We've come to the last digit in the original number which was 5, we had a remainder of 6 from the previous part and so changed the number to 65, divided it by 7 and the result has a remainder of 2, which means that it is not exactly divisible by 7.

Honestly, this is primary school maths, I remember being taught this years ago. However it only shows if a number is directly divisible by 7, the remainder we have when we've finished the process only tells us that it isn't directly divisible by whatever number we were dividing by originally.

If you haven't heard of long division, it's something you can pick up and learn very easily on the internet, and although I imagine you won't be dividing massive numbers very often, it's still worth learning.

Also, the answer comes to

Twelve thousand, five hundred, sixty nine and two sevenths 12569 + 2/7

I dont know what is the right solution is and I just do some of my own techniques. On this problem, I just simply add all the numbers in the dividends. 87985 = 8+7+9+8+5 = 37 Then simply identify if the sum of the dividends is divisible by 7. Obviously, 37 is not divisible by 7. So, the answer is simply NO.

Finding a number can be divided by seven : divisibility rule of 7 The divisibility rule for number 7 helps in finding whether a number is exactly divisible by 7. The rules are illustrated with clear examples for easier understanding. Divisbility Rule of 7 : The last digit is multiplied by 2 and subtracted from the rest of the number. The result is either 0 or divisible by 7.

Example : i) 2205 ii) 9751

Explanation : i) 2205 Last digit multiply by 2, 5 * 2 = 10 Subtracted rest of digits, 220 - 10 = 210 Divisible by 7, 210 / 7 = 30

ii) 9751 Last digit multiply by 2, 1 * 2 = 2 Subtracted rest of digits, 975 - 2 = 973 Last digit multiply by 2, 3 * 2 = 6 Subtracted rest of digits, 97 - 6 = 91 Last digit multiply by 2, 1 * 2 = 2 Subtracted rest of digits, 9 - 2 = 7 Divisible by 7, 7 / 7 = 1

Both the numbers can be divided by 7.

One way to determine the number's divisibility by 7 is to subtract large numbers known to be divisible by 7. Choose numbers that can be subtracted easily using mental math. Eventually, this will yield a number that is small enough to recognize easily as being divisible by 7 or not.

87985 - 84000 = 3985

3985 - 3500 = 485

485 - 420 = 65

65 isn't divisible by 7, so 87,958 is not divisible by 7.

70,000 is divisible by 7 --> 87,985 - 70,000 = 17,985 ... 14,000 is divisible by 7 --> 17,985 - 14,000 = 3,985 ... 3,500 is divisible by 7 --> 3,985 - 3,500 = 485 ... 420 is divisible by 7 --> 485 - 420 = 65 ... 65 is not

I just did it the old fashion way in my head and ended up with 65 and immediately knew 7 doesn't divide cleanly into that. it would be 9 with a remainder of 2. 7 / 87985 = 12569 with a remainder of 2

87985 : 7

879 85

85 is not Divisible by 7

LOL

8+7+9+8+5=37 is nt divided by 7

LOL. Now I know that there is a technique. All this time, i take the last 2 digits and divide it whether it is divisible or not :)

Any number is divisible by another number..Just may not be a whole number

8 - 7 = 1 then 17 - 14 = 3 then 39 - 35 = 4 then 48 - 42 = 6 then 65 is clearly not divisible by 7..

I know 87500 is a multiple of 7. The remainder 485 difers by 5 from 490 another multiple of 9. Then 87985 is not divisible by 7

A number is divisible by 7=>A last two digit divisible by 7

i don't know if my trick is 100% true :D

sizeof "87985" is 5 --> odd and 7 is --> odd,

odd/odd = no :D

even/odd = may-be :v

8+7+9+8+5 = 37 37 % 7 = 2 since remainder is 2 , so 87985 is not divisible by 7.