QUANTUM COMPUTING - EXAMPLE 32.3 : A system of linear congruences consists of 3 equations : X ≡ 1 (mod 3), X ≡ 3 (mod 5), X ≡ 4 (mod 6). X has positive values. (a) List the values of these equations from 1 to 35. Then find the minimum value of X. (b)(i) Find the least common multiple (LCM) of b = 3, 5 and 6 where X ≡ a (mod b). (ii) If b - a has the same value of all equations above, then X + (b - a) is divisible by LCM. Find the value of minimum value of X via LCM division.
QUANTUM COMPUTING - ANSWER 32.3 : (a) X ≡ 1 (mod 3) = 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34. X ≡ 3 (mod 5) = 3, 8, 13, 18, 23, 28, 33. X ≡ 4 (mod 6) = 4, 10, 16, 22, 28, 34. All equations have minimum value of X = 28. (b)(i) LCM for b = 3, 5 and 6 = (30 / 10, 30 / 6, 30 / 5) is 30. (ii) Since b - a = 3 - 1 = 5 - 3 = 6 - 4 = 2, then X + (b - a) is divisible by LCM. X + 2 is divisible by 30. X = 30 - 2 = 28. The answer is given by Kang Chuen Tat; PO Box 6263, Dandenong, Victoria VIC 3175, Australia; SMS +61405421706; chuentat@hotmail.com; http://kangchuentat.wordpress.com.
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UNIT OPERATION - EXAMPLE 9.2 : A distillation column separates 10000 kg / hr of a mixture containing equal mass of benzene and toluene. The product D recovered from the condenser at the top of the column contains 95 % benzene, and the bottom W from the column contains 96 % toluene. The vapor V entering the condenser from the top of the column is 8000 kg / hr. A portion of the product from the condenser is returned to the column as reflux R, and the rest is withdrawn as the final product D. Assume that V, R, and D are identical in composition since V is condensed completely. Find the ratio of the amount refluxed R to the product withdrawn D. Hint : Solve the simultaneous equations as follow in order to find the answer (R / D) : 10000 = D + W; 10000 (0.5) = D (0.95) + W (0.04); 8000 = R + D.
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