answer:Joseph traveled to half the number of countries George traveled to, so Joseph traveled to 6/2 = 3 countries. Patrick traveled to three times the number of countries Joseph traveled to, so Patrick traveled to 3*3 = 9 countries. Zack has traveled to twice the number of countries Patrick traveled to, so Zack traveled to 9*2 = 18 countries. Therefore, Zack traveled to 18 countries. 18 The answer is: 18

answer:To solve this problem, we need to determine the value of x, which represents the number of tree huggers at the party. Let's break down the information given: Number of poets: x + 50 (50 more than tree huggers) Number of singers: 110 Total number of students: 400 We can set up the equation as follows: Number of poets + Number of singers + Number of tree huggers = Total number of students (x + 50) + 110 + x = 400 Let's simplify and solve for x: 2x + 160 = 400 To isolate x, we subtract 160 from both sides of the equation: 2x + 160 - 160 = 400 - 160 2x = 240 To solve for x, we divide both sides of the equation by 2: x = 240 / 2 x = 120 The value of x is 120. 120 The answer is: 120

answer:In the first scenario, let x be the number of Dummies in each bag. Since the Dummies cannot be divided equally among 9 kids, we know that x is not divisible by 9. We also know that there are 7 pieces left over after each kid receives an equal number of pieces. Therefore, x must be congruent to 7 modulo 9. In the second scenario, we have 3 bags, so the total number of Dummies is 3x. We want to divide these Dummies equally among 9 kids. Let y be the number of Dummies each kid receives. We want to find the remainder when 3x is divided by 9. Since x is not divisible by 9, we know that 3x is also not divisible by 9. Therefore, 3x leaves a remainder when divided by 9. Let r be this remainder. We want to find the smallest possible value of r. Since x is congruent to 7 modulo 9, we have 3x equiv 3 cdot 7 equiv 21 equiv 3 pmod{9}. Therefore, the remainder r is boxed{3}. The answer is: 3

answer:We want to find the value of X in the given situation. To find the product of the digits in the base X representation of 6543_{10}, we need to convert the number 6543 from base 10 to base X. To do this, we can divide 6543 by powers of X and keep track of the remainders. Starting with the largest power of X, we divide 6543 by X^3: frac{6543}{X^3} = a_3 Next, we divide the remainder by the next power of X, X^2: frac{a_3}{X^2} = a_2 We repeat this process until we divide the remainder by X^0 = 1: frac{a_1}{X^1} = a_0 To find the value of X, we need to determine the number of digits in the base X representation of 6543. Since the product of the digits in the base X representation is 168, we can write: 1 cdot 6 cdot 8 = 168 We can see that the digits 1, 6, and 8 are all less than X. So, the value of X must be greater than 8. Since X represents the base of the number system, it must be a positive integer. Therefore, X can be any positive integer greater than 8. The answer is: 8