I agree with the other person. <u>The answer is 5.</u>
Here's how we can check to see if a number is divisible by 7.
- Step 1) Remove the units digit to form a truncated (aka shrunken) number.
- Step 2) Double the digit we removed.
- Step 3) Subtract that result from the truncated number.
If the result of the third step is a multiple of 7, then so is the original value itself. We can repeat the steps to progressively shrink the values down so we have something more manageable.
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Let's use those steps to see if 20,258 is a multiple of 7
- Step 1) Remove the '8' from 20258 to to get 2025
- Step 2) Double the 8 to get 16
- Step 3) Subtract 16 from 2025 to get 2025 - 16 = 2009
It's not clear if 2009 is a multiple of 7 or not; however, we can apply the same three steps to shrink the number down. This process can be repeated as many times as you like.
- Step 1) Remove the 9 from 2009 to get 200
- Step 2) Double the 9 to get 18
- Step 3) Subtract to get 200-18 = 182
Like before, the result 182 is fairly big and it's not clear if we have a multiple of 7. Luckily, we only need to do one more round of those steps.
- Step 1) Remove the 2 from 182 to get 18
- Step 2) Double that value we removed to get 4
- Step 3) Subtract to get 18-4 = 14
The result 14 is a multiple of 7, which means that 182 is also multiple of 7. By a domino-like chain reaction, the values 2009 and 20258 are also multiples of 7.
Notes:
- 182 = 26*7
- 2009 = 287*7
- 20258 = 2894*7
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We've just verified that 20258 is a multiple of 7. Let's go over an example a non-multiple of 7. I'll add 1 to the number 20258 so we are guaranteed that it cannot possibly be a multiple of 7. Doing the round of steps as shown above should get us to some smaller non-multiple of 7.
- Step 1) Remove the 9 from 20259 to get 2025
- Step 2) Double the 9 to get 18
- Step 3) Subtract: 2025 - 18 = 2007
Repeat the steps until we get something more manageable
- Step 1) Remove the 7 from 2007 to get 200
- Step 2) Double the removed digit: 7*2 = 14
- Step 3) Subtract: 200-14 = 186
Let's do one more round
- Step 1) Remove the 6 from 186 to get 18
- Step 2) Double the digit removed: 2*6 = 12
- Step 3) Subtract: 18-12 = 6
The value 6 is not a multiple of 7, and neither are 186 and 2007 and 20259.
If you have your calculator with you, then you should see that dividing any of those values over 7 will not lead to a whole number result.
- 186/7=26.57
- 2007/7=286.71
- 20259/7=2894.14
This is a fairly quick and efficient way to check divisibility by 7 if you are in a hurry.
