### Brain Teasers

# Ten Numbers

There are ten natural numbers, two of which are equal. Adding any nine of them, we can get the following sums:

82, 83, 84, 85, 87, 89, 90, 91, 92.

Find the largest number of the ten.

82, 83, 84, 85, 87, 89, 90, 91, 92.

Find the largest number of the ten.

### Answer

There are ten possible sums, arrived at by alternately excluding each one of the ten numbers. Two of the sums are equal. If we add the ten sums, it should equal nine times the sum of the ten numbers, as each number will be present within all of the sums nine times.82+83+84+85+87+89+90+91+92=87*9=783, which is divisible by 9, therefore the repeated sum must be divisible by 9. The only sum which meets this criteria is 90.

The sum of the ten numbers is (783+90)/9=97. The largest number is found by subtracting the lowest sum of nine numbers (which excluded the largest of the ten numbers) from the sum of all ten numbers. Thus: 97-82=15. 15 is the largest number of the ten.

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