MHB Smallest Number Divisible by 99 Using Digits 1-9

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Find the smallest number that is made up of each of the digits 1 through 9 exactly once and is divisible by 99.
 
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kaliprasad said:
Find the smallest number that is made up of each of the digits 1 through 9 exactly once and is divisible by 99.
my solution:
the smallest number that is made up of each of the digits 1 through 9 exactly once and is divisible by 99 is:
123475869
for 1+2+3+4+7+5+8+6+9=45 mod 9=0
2+4+5+6=17
1+3+7+8+9=28
28-17=11 mod 11=0
the largest number that is made up of each of the digits 1 through 9 exactly once and is divisible by 99 is:
987652413
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...

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