Number Theory:PRMO
Number Theory
Level
2
The equation 166*56=8590 is valid in some base b>10 or b=10{that is, 1,6,5,8,9,0 are digits in base b in the above equation}. Find the sum of all possible values of b>10 or b=10 satisfying the equation.
10
5
12
9
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1 solution
This isn't true in base 10 and two of the choices can be ruled out because the problem specifies b>10. That leaves only 1 2
Not to mention the digit 9 only exists in bases above 9.
Looking at the last digits 6 × 6 = 0 can only occur in base 9,12,18,36.
Finally, the equation ( b 2 + 6 b + 6 ) ( 5 b + b ) = 8 b 3 + 5 b 2 + 9 b has only a singe real solution b = 1 2