Finding the Remainder of 111111222222 Divided by 7

[buzzmath]

Can anyone help me? I'm trying to find the remainder when 111111222222 is divided by 7 without using any long division.

I thinking that i can take the alternating sums of the 3 digits blocks and set that mod7. I'm not sure if I'm on the right track or not.
thanks

 

[Tide]

Can anyone help me? I'm trying to find the remainder when 111111222222 is divided by 7 without using any long division.

I thinking that i can take the alternating sums of the 3 digits blocks and set that mod7. I'm not sure if I'm on the right track or not.
thanks

That will work since 1001 mod 7 = 0.

 

[AKG]

I guess the first thing to do is use a calculator and actually figure out the answer. Then do the question making the right factorizations that help you prove the answer you compute, where these factorizations should be the kind you can see without a calculator

111111222222
= (111000222)(1001) since you can easily see this without a calculator

(1001)
= 10³ + 1

Figure out from here how to show 1001 = 0 (mod 7) [Hint: what is 10 (mod 7)?]

 

[HallsofIvy]

Or you could use a divisibility test. Break the number into pairs of digits:
11 11 11 22 22 22. Now, starting at the left, write the difference between the first pair and the next larger or equal multiple of 7. Since the first pair is 11 and the next larger multiple of 7 is 14, that's 3. Then the difference between that pair and the next smaller or equal multiple of 7. Since the next pair is 11, 11- 7= 4. Alternate between "larger" and "smaller" multiples of 7 as you go down the pairs. For this number, 111111222222, you get 3, 4, 3, 6, 1, 6. Write the digits in reverse order: 616343. Now repeat the process until you get something that obviously is or is not a multiple of 7. The original number is a multiple of 7 if and only if that first number is.

(Of course, I'm giving away the answer just by saying that a divisibility test WILL work!)

 

[VietDao29]

Or you could use a divisibility test. Break the number into pairs of digits:
11 11 11 22 22 22. Now, starting at the left, write the difference between the first pair and the next larger or equal multiple of 7. Since the first pair is 11 and the next larger multiple of 7 is 14, that's 3. Then the difference between that pair and the next smaller or equal multiple of 7. Since the next pair is 11, 11- 7= 4. Alternate between "larger" and "smaller" multiples of 7 as you go down the pairs. For this number, 111111222222, you get 3, 4, 3, 6, 1, 6. Write the digits in reverse order: 616343. Now repeat the process until you get something that obviously is or is not a multiple of 7. The original number is a multiple of 7 if and only if that first number is.

(Of course, I'm giving away the answer just by saying that a divisibility test WILL work!)

This is new, I have never heard of it. But what if the number of digits is an odd number. How can one split it into pairs? Say, the number is: 12345?