There are eleven three-digit multiples of 9 that consist only of odd digits.
To find these multiples, we need to consider three-digit numbers formed using the digits 1, 3, 5, 7, and 9, and then check which of them are divisible by 9. A number is divisible by 9 if the sum of its digits is divisible by 9.
Let's analyze the possible combinations of odd digits:
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Sum of digits equals 9: The only combination is 1 + 3 + 5. This gives us the following numbers: 135, 153, 315, 351, 513, and 531.
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Sum of digits equals 18: The possible combinations are 1 + 7 + X (no odd single digit X exists that will make the sum 10) and 3 + 5 + X (the odd single digit X must be equal to 10). The combinations 5 + 5 + X (the odd single digit X must be equal to 8) and 7 + 1 + X (no odd single digit X exists that will make the sum 10) and 9 + X + X (where X is 9/2 = 4.5, and that doesn't exist) will result in a double digit number that won't work either.
The other possibility is 9 which can be obtained using 1+8+9,2+7+9,3+6+9,4+5+9,3+7+8, 3+8+7. Using only the odd digits 1+3+5 which are = 9 and using 1+3+5 twice: we can get 1+1+7, 1+3+5, 3+3+3, 5+1+3. -
Sum of digits equals 27: To obtain the sum of 27, we need three identical digits that add up to 27 which is achieved when 9+9+9 since the only odd digit is 9. However there are no other combinations of odd digits which will add to 27.
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Numbers formed using three 3's: 3 + 3 + 3 = 9. The only possible number is 333.
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Numbers formed using three 9's: 9 + 9 + 9 = 27. The only possible number is 999.
Combining these, we have the following multiples of 9 consisting only of odd digits:
- 135, 153, 315, 351, 513, 531
- 333
- 555
- 999
- 117, 171, 711
- 575, 755, 557
In summary, they are: 135, 153, 315, 351, 513, 531, 333, 555, 999, 117, 171, 711, 575, 755, 557.
Adding up all the combinations we have 6+1=7 odd numbers that are the multiples of 9.
Considering only the odd digits 1, 3, 5, 7 and 9:
- 1+1+7=9
- 3+3+3=9
- 1+3+5=9
- 9+9+9=27
- 1+5+3=9
- 5+5+7=27
- 7+1+1=9
- 3+1+5=9
- 3+5+1=9
- 5+1+3=9
- 5+3+1=9
Therefore, the complete list is: 135, 153, 315, 351, 513, 531, 333, 117, 171, 711, 999, 555, 557, 575 and 755.
Therefore there are eleven three-digit multiples of 9 consisting only of odd digits.
