For non-negative integers a and b , Let T i denote the i th largest positive integer such that there's no solution to 1 1 a + 1 2 b = T i .
What is the value of T 1 + T 2 + T 3 + T 4 + T 5 + T 6 + T 7 + T 8 + T 9 + T 1 0 ?
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How did you get these numbers?
All numbers except 11 12 22 23 24 and so on. 11n ...12... excluded for each n.
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How do you know this is true? How do you know you have accounted for every single integer?
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11 and 12 are relative primes, 11n+12m=1 has solution n=11 m=-10 so we can solve for any T, but for some numbers a and b will be negative. For large positives a and be will be also positive.
Every time exclude one more number and at the end we can get any number >=110
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Exclude one more number from what?
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Numbers are 1 2 3 4 5 6 7 8 9 10 13 14 15 16 17 18 19 20 21 25 26 27 28 29 30 31 32 37 38 39 40 41 42 43 49 50 51 52 53 54 61 62 63 64 65 73+74+75+76+85+86+87+97+98+109