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Question 576742: Consider the whole number with one thousand digits that can be formed by writing four digits, 2772, two-hundred fifty times in succession.
Is this number divisible by 9? Why or why not?
Is this number divisible by 11? Why or why not?
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! 2772 is divisible by 9 because 2+7+7+2 = 18, which is divisible by 9. If we write 2772 250 times in a row, the digit sum will be 250(18), also a multiple of 9.
2772 is also divisible by 11 because the difference between the sum of the even-positioned digits and the odd-positioned digits is a multiple of 11 (if you know the divisibility rule for 11). Writing it 250 times in a row will not change the digit sum. A better way to think of it is, 277227722772...2772 is equal to 2772 times some number (100010001...1). Since 2772 is divisible by 11, the number 27722772...2772 will also be divisible by 11.
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