answer:Let M = 123456789012. We need to find the remainder of M when divided by 210 = 2 times 3 times 5 times 7. We will use the Chinese Remainder Theorem again. 1. **Divisibility by 2**: Since the last digit of M is 2, M is even. Hence, M equiv 0 pmod{2}. 2. **Divisibility by 3**: To check divisibility by 3, sum the digits of M: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 0 + 1 + 2 = 48. Since 48 is divisible by 3, M equiv 0 pmod{3}. 3. **Divisibility by 5**: The last digit of M is 2, which is not divisible by 5. Therefore, M equiv 2 pmod{5}. 4. **Divisibility by 7**: Calculating M modulo 7 directly can be simplified by taking the last few digits or applying specific divisibility tricks. Using direct calculation, M mod 7 = 123456789012 mod 7 = 3 (e.g., by reducing the number in parts or calculator usage). Using the Chinese Remainder Theorem: - From the relations M equiv 0 pmod{2}, M equiv 0 pmod{3}, M equiv 2 pmod{5}, and M equiv 3 pmod{7}, we need to find a number x that satisfies all these congruences. - We find x that satisfies x equiv 2 pmod{5} and x equiv 3 pmod{7}. By trial or systematic application of the extended Euclidean algorithm: - Testing candidates based on 5k + 2 for k = 0, 1, 2, dots, we find x = 17 fits both conditions (17 equiv 3 pmod{7}). - Since x = 17 also satisfies x equiv 0 pmod{6} (as both 2 and 3 divide 0), hence it is a solution. Thus, M equiv boxed{17} pmod{210}.

answer:The scenario described is that of a dog tied to a tree with a 10-foot cord, running around the tree with the cord fully extended. If the dog ran approximately 30 feet, it means the dog has run around the circumference of a circle with the tree at the center and the cord as the radius. The circumference (C) of a circle is given by the formula: C = 2πr where r is the radius of the circle. In this case, the radius (r) is the length of the cord, which is 10 feet. Plugging this into the formula gives us: C = 2π(10 feet) = 20π feet The actual distance the dog ran is approximately 30 feet, which is less than the full circumference of the circle (20π feet is approximately 62.83 feet). This means the dog did not run a full circle around the tree. Since the dog ran approximately 30 feet, it ran less than half the circumference of the circle. This indicates that the dog started running from one side of the tree and continued to the opposite side, making approximately a half-circle around the tree. Therefore, the direction from which the dog started running cannot be determined with the information given, as the dog could have started running from any point around the tree and still covered approximately boxed{30} feet to reach the opposite side. The direction is relative to an observer's position or a given reference point, which is not provided in the scenario.

answer:Let's denote the number of miles Mona biked on Monday as M, on Wednesday as W, and on Saturday as S. According to the information given: 1. Mona biked a total of 30 miles for the week: M + W + S = 30 2. On one of the days, she biked 6 miles. Without loss of generality, let's assume that's the distance she biked on Monday: M = 6 3. On Saturday, she biked twice as far as on Monday: S = 2M Now we can substitute the values we know into the equations: M = 6 (from assumption) S = 2M = 2 * 6 = 12 (from the third point) Now we can find out how many miles she biked on Wednesday by substituting M and S into the first equation: M + W + S = 30 6 + W + 12 = 30 W = 30 - 6 - 12 W = 30 - 18 W = 12 So, Mona biked boxed{12} miles on Wednesday.

answer:**Analysis** This problem tests the understanding of the definition, standard equation, and simple properties of a parabola. It is a moderately difficult question. **Solution** The standard equation of the parabola is y^2 = 8x, with p = 4, and the focus is F(0, 2). The equation of the directrix is y = -2. Let d represent the distance from point P to the directrix. Then, PF = d. Therefore, finding the minimum value of PA + PF is equivalent to finding the minimum value of PA + d. Clearly, drawing a perpendicular line AQ from point A to the directrix y = -2, when point P is the intersection of line AQ and the parabola, PA + d reaches its minimum value. The minimum value is AQ = 5 - (-2) = 7. boxed{text{Final answer is } 7.}