The TEN DIGIT Teaser
It's a simple enough question:
Can you form a ten-digit number using each digit once so that...
- the first digit on its own divides by 1
- the first two digits on their own divide by 2
- the first three digits on their own divide by 3
- the first four digits on their own divide by 4
... and so on until...
- the first nine digits on their own divide by 9
- the complete ten digit number divides by 10?
?
Suppose your answer was 7234619850
7 divides by 1, so that's ok.
72 divides by 2
723 divides by 3
7234 ... doesn't divide by 4, so it's no good!
The really neat thing is that there is only ONE possible answer!
To do this you'll need the dividing tests.
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When does a number divide by 1-10? |
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1/ | Anything divides by 1
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2/ | Number must be even (ending 2,4,6,8,0)
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3/ | Add the digits in the number. If the answer divides by 3, so does the number.
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4/ | The last two digits must divide by 4. To test: if the second-last digit is odd, the last digit must be 2 or 6. If the second-last digit is even, the last digit must be 0,4 or 8.
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5/ | Last digit must be 0 or 5.
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6/ | Number must be even and divide by 3.
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7/ | Remove last digit, and multiply it by 2. Subtract from other digits. Answer must be 0 or divide by 7. E.g. 623: remove 3 and x2 = 6. 62-6 = 56 which divides by 7, so does 623.
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8/ | The last three digits must divide by 8. To test: if the third-last digit is even, the last two digits must divide by 8. If the third-last last digit is odd, last two digits must divide by 4, but NOT 8.
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9/ | Add the digits in the number. If the answer divides by 9, so does the number.
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10/ | There's a 0 on the end.
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Stuck? Then click on our Pure Mathematicians for the answer!
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