Question 1108922: The number 111...111000...000, made of 39 ones followed by 93 zeros, is NOT divisible by
Possible answers : a)12 b)16 c)22 d)37 e)625
Found 2 solutions by Alan3354, ikleyn:
Answer by Alan3354(69443)   (Show Source):
Answer by ikleyn(53762)  (Show Source):
You can put this solution on YOUR website! .
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This problem deserves more explanations.
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a) Obviously, the given number is divisible by 4 and by 3, so it is divisible by 12.
For divisibility on 3, apply the divisibility by 3 rule:
The number is divisible by 3 if and only if the sum of its digits is divisible by 3.
See the lesson Divisibility by 3 rule in this site.
It is the case, since the sum of the digits 39 is divisible by 3, the number is divisible by 3.
b) The number 10000 is divisible by 16 (direct check !),
Hence, the given number is divisible by 16.
c) The given number IS NOT divisible by 11.
For divisibility on 11, apply the divisibility by 11 rule:
The number is divisible by 1 if and only if the alternate sum of its digits is divisible by 11.
See the lesson Divisibility by 11 rule in this site.
It is the case, the alternate sum of the digits is 1 (one). Since it is not divisible by 11, the number itself IS NOT divisible by 11.
Since the number is not divisible by 11, it IS NOT divisible by 22.
d) Regarding the number 37, it is useful to know that 111 = 37*3.
It follows from this fact that the given number is divisible by 37.
e) 625 = 5^4. 10^4 = 10000 is divisible by 5^.
So the given number is divisible by 625.
From this considerations, the only answer to the problem's question is the number 22.
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