Divisible by 17?

Question

Which one is divisible by 17?

355687428096000

or

355687428096800

Our teacher told us that it can be solved in 10 seconds.

Solve it without using a caluculator.

Answer 1

The second choice because you will have to multiply 17 by multiples of 100 to get zeroes in the last two places. 17 will go into 68 but never into 60.

That reasoning rules out the first number but doesn't guarantee that 17 will go into the second... so there may be more to this that I'm not seeing at a 10 second glance.

Aloha

P.S. Pascal is right... my glance was wrong. 17 goes into the first number. So we are still left wondering what the 10 second solution is?

Answer 2

Following the pattern of taking the last digit, multiplying it by 5, then subtracting it from the remaining digits, doing this successively until you get a remainder of 0 or a known multiple of 17. (See the attached website for details.)

35568742809 6 6 * 5 = 30
-30
--------
3556874277 9 9 * 5 = 45
-45
------------
355687423 2 2 * 5 = 10
-10
----------
35568741 3 3 * 5 = 15
-15
-----------
3556872 6 6 * 5 = 30
-30
--------
355684 2 2 * 5 = 10
-10
---------
35567 4 4 * 5 = 20
-20
----------
3554 7 7 * 5 = 35
-35
------
351 9 9 * 5 = 45
-45
----
30 6 6 * 5 = 30
-30
----
0


Therefore the first number is divisible by 17.

Answer 3

Any number that is divisible by 17 will have this property: multiply the last digit by 5, then subtract that number from the remaining digits and the resulting number will be divisible by 17. But that takes way more than 10 seconds. Good luck!

Answer 4

Who are the absolute morons who gave a thumbs-up to enufwork's answer? It's WRONG - the first number is divisible by 17 (specifically, it is equal to 20922789888000*17 - try it out on your calculator). Conversely, the second number is not divisible by 17. Now, I may not know what the correct answer is, but I find it to be intellectually abhorrent not to even bother to check the answer before approving it. A thumbs-down to the lot of you.