Divisible by 14?
How many of the following octal (base 8) numbers are divisible by 1 4 1 0 (decimal)?
-
A. 3 4 6 0 3 4 3 6 1 0 7 5 3 6 2 5 2 3 6 5 0 2 3 5 6 4 2 0 3 3 5 4 3 5 3 5 0 1 1 3 5 3 4 5 4 4 2 8
-
B. 2 3 4 2 5 3 4 0 7 4 5 6 0 2 3 4 2 5 4 6 0 7 4 5 2 0 1 3 4 3 6 3 2 5 2 5 5 6 7 7 0 4 3 5 2 3 5 0 6 4 5 8
-
C. 2 3 5 7 6 7 3 4 2 3 5 0 2 3 5 6 3 4 5 2 3 4 0 5 2 3 4 0 2 5 3 0 5 2 3 5 0 0 5 3 4 5 2 0 0 0 5 3 5 0 0 0 8
-
D. 2 3 0 4 3 0 4 6 7 0 2 0 5 0 4 7 5 0 0 1 0 0 3 3 6 3 6 0 7 0 7 0 5 0 2 0 3 5 3 4 5 6 4 3 0 5 4 6 5 0 3 4 5 6 8
-
E. 3 5 7 4 5 6 0 4 5 6 3 5 4 2 3 5 0 4 6 5 4 6 6 5 7 0 4 3 5 5 4 3 5 0 6 5 7 3 4 5 0 4 5 5 0 6 7 8
-
F. 4 7 0 2 3 4 5 0 3 4 6 2 0 3 4 5 3 2 5 0 4 7 6 6 5 0 5 7 5 0 6 4 0 5 7 7 5 0 6 4 0 3 4 6 0 6 7 5 7 0 8
-
G. 5 4 7 0 4 5 2 5 4 6 0 4 6 7 0 6 7 5 6 0 6 3 5 5 3 4 6 0 4 0 5 3 2 0 1 0 1 4 2 3 4 0 1 5 2 3 0 5 2 1 5 4 8
-
H. 3 2 6 0 4 3 5 0 3 6 4 5 7 5 4 6 4 3 2 4 3 4 6 4 6 0 4 7 5 4 0 5 4 2 5 3 4 6 6 4 5 0 3 2 4 5 6 0 4 8
-
I. 1 4 6 7 0 5 4 5 7 0 7 0 7 0 4 5 6 7 7 0 0 6 4 5 6 7 0 4 5 0 6 4 5 6 0 4 2 1 0 2 3 4 1 2 3 5 1 5 0 3 2 8
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
I'll leave the proof to others, but here is the trick to solve this problem. 14 = 2 * 7, so the number is divisible by 14 if it is divisible by 2 and by 7.
i. 2 divides 8, so an octal number is divisible by 2 if the last digit is even (like with base 10).
ii. In base 10, a number is divisible by 9 if the sum of the digits of the number is divisible by 9; likewise, in base 8, a number is divisible by 7 if the sum of the digits is divisible by 7. If you sum octally, you can keep adding the digits, and if the result is 0 or 7, the number is divisible by 7, otherwise not. E.g. the sum of E is 196 (decimal), that is 304 (octal), and 3 + 4 = 7.
The (decimal) digitsums are resp.: 159, 185, 155, 166, 196, 189, 172, 187 and 178. 196 and 189 are divisible by 7, but E is an odd number; only F is divisible by 2 and 7, and therefore by 14.