10^25 – 560 is divisible by all of the following EXCEPT:

BTGmoderatorDC wrote:10^25 – 560 is divisible by all of the following EXCEPT:

A. 11
B. 8
C. 5
D. 4
E. 3

OA E

Source: Manhattan Prep

We see that 10^25 is divisible by 8, 5 and 4 and so is 560, therefore, the difference, 10^25 - 560, is also divisible by 8, 5 and 4. This leaves us to check 11 and 3.

Notice that when a number is subtracted from 10^n and suppose that number is at least 2 digits fewer than 10^n, then the difference will have a sequence of leading 9’s. For example, if 56 is subtracted from 10^5, the difference is 100,000 - 56 = 99,944. Furthermore, we see that when subtracting 56 from 10^5, we just need to use 100 to subtract 56 (since 100 is greater than 56 already). Notice that 100 - 56 = 44 and the remaining digits (to the left of 44) will be a sequence of leading 9’s. Let’s extend this concept to 10^25 - 560.

We see that we need to use 1000 to subtract 560 (since 1000 is greater than 560 already). Notice that 1000 - 560 = 440 and the remaining digits (to the left of 440) will be a sequence of leading 9’s. That sequence of 9’s is divisible by 3; however, 440 isn’t. Therefore, the difference, 10^25 - 440 = 99...9440 (we don’t need to care how many 9’s there are), will not be divisible by 3.

(Note: We will leave as an exercise for readers to show that 10^25 - 560 is divisible by 11. Hint: Argue that 440 in 99...9440 is divisible by 11 and that the sequence of 9’s has an even number of 9’s.)

Answer: E