# I am a number greater than 40,000 and less than 60,000. My ones digit and tens digit are the same. My ten-thousands digit is 1 less than 3 times the sum of my ones digit and tens digit. My thousands digit is half my hundreds digit, and the sum of those two digits is 9. What number am I?

I am a number greater than 40,000 and less than 60,000:
40,000 < n < 60,000
This means that:
n = 10,000n₁ + 1,000n₂ + 100n₃ + 11n₄
And also:
4 ≤ n₁ < 6
0 ≤ n₂ ≤ 9
0 ≤ n₃ ≤ 9
0 ≤ n₄ ≤ 9
My ten thousands digit is 1 less than 3 times the sum of my ones digit and tens digit:
n₁ = 3*2n₄ - 1
n₁ = 6n₄ - 1
This means that:
n = 10,000*(6n₄-1) + 1,000n₂ + 100n₃ + 11n₄
n = 60,000n₄ - 10,000 + 1,000n₂ + 100n₃ + 11n₄
n = 60,011n₄ - 10,000 + 1,000n₂ + 100n₃
My thousands digit is half my hundreds digit, and the sum of those two digits is 9:
n₂ = 1/2 * n₃
n₂ + n₃ = 9
Therefore:
n₂ = 9 - n₃
Therefore:
9 - n₃ = 1/2 * n₃
9 = 1/2 * n₃ + n₃
9 = 1.5 * n₃
Therefore:
n₃ = 6
If n₃=6, n₂=3.
This means that:
n = 60,011n₄ - 10,000 + 1,000*3 + 100*6
n = 60,011n₄ - 10,000 + 3,000 + 600
n = 60,011n₄ - 6,400
Therefore:
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