How many seven-digit numbers are there that are divisible by 11?
In other words, I want to find the number of seven digit numbers that are divisible by 11.
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Well, the last number divisible by 11 before 7 digits is 999999, so the answer is $$\left\lfloor \frac{9999999-999999}{11}\right\rfloor = 818181.$$ More generally, if $a$ is divisible by 11 and $b>a$, then the number of integers in the half open interval $(a, b]$ divisible by $11$ is given by $\left\lfloor \frac{b-a}{11} \right\rfloor$, as you can verify by trying a few small cases, and then an induction argument should follow pretty easily.
Dustan Levenstein
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many thanks for your answer but i want the proof of the theorem or text that you write for number of 7 digit number are divisible by 11 ? please help me :( – babak Nov 28 '13 at 22:38
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Have you made any attempt at proving the theorem? – Dustan Levenstein Nov 29 '13 at 03:10
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no i do not have any idea for proving – babak Nov 29 '13 at 11:04
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Try taking $a=0$ or $a=11$, and look at some small values of $b$, say $b$ up to $30$ or $40$ or so. You should see a pattern form fairly easily. Are you aware that $\lfloor \cdot \rfloor$ represents the floor function? – Dustan Levenstein Nov 29 '13 at 13:24
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many thanks my mind was very busy because of that ... ok i know the reason .bye – babak Nov 29 '13 at 19:44
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90909 * 11 is the smallest number 7 digit number divisible by 11 909090 * 11 is the largest 7 digit number divibsle by 11
So there are precisely 818181 such numbers
kjata
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