Why is any no. Repeated 6 times always divisible by 37?
Answers
9 + 138 + 431 + 772 = 1350
1 + 350 = 351
351 differs from 333 by 18
18 is not a multiple of 37
Therefore 9,138,431,772 is not a multiple of 37
4 + 48 + 675 + 309 = 1036
1 + 36 = 37
37 is a multiple of 37
Therefore 4,048,675,309 is a multiple of 37
Suppose you have a number D with four or more digits. This number can be written as 1000X + 100c + 10b + a, where a, b, and c are the last three (lowest order) digits, and X is the number formed by all the rest of the digits. For convenience of notation, let cba = 100c + 10b + a, not to be confused with c×b×a. In other words, D = 1000X + cba.
Since 999X is divisible by 37, subtracting it from D will not change the divisibility; i.e., D is equivalent to D - 999X, modulo 37. But D - 999X = X + cba, so this is just the same as adding the "front matter" onto the last three digits.
This doesn't quite show that the number can be broken completely down into three-digit parts before adding them all together, but if you've read this far without keeling over from boredom or confusion, then
(1) you can probably see that it would generalize; and
(2) you're probably capable of doing a more rigorous mathematical proof of the whole shebang than I am anyway.
Answer:
inbox please
Step-by-step explanation:
...........................