Diffusion

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question:"Modulo m graph paper" consists of a grid of m^2 points, representing all pairs of integer residues (x,y) where 0le x, y <m. To graph a congruence on modulo m graph paper, we mark every point (x,y) that satisfies the congruence. For example, a graph of yequiv x^2pmod 5 would consist of the points (0,0), (1,1), (2,4), (3,4), and (4,1). The graph of 3xequiv 4y-1 pmod{35}has a single x-intercept (x_0,0) and a single y-intercept (0,y_0), where 0le x_0,y_0<35. What is the value of x_0+y_0?

answer:To find the x-intercept, we set y to 0 and solve for x: 3xequiv 4(0)-1pmod{35} 3xequiv -1pmod{35} Since 3 and 35 are relatively prime, we can multiply both sides by the modular inverse of 3 modulo 35 to find x: 3^{-1}(3x)equiv 3^{-1}(-1)pmod{35} xequiv -3^{-1}pmod{35} Since 3^{-1}equiv 12pmod{35}, we have xequiv -12equiv 23pmod{35}. Therefore, the x-intercept is (23,0). To find the y-intercept, we set x to 0 and solve for y: 3(0)equiv 4y-1pmod{35} 0equiv 4y-1pmod{35} Since 4 and 35 are relatively prime, we can multiply both sides by the modular inverse of 4 modulo 35 to find y: 4^{-1}(0)equiv 4^{-1}(4y-1)pmod{35} 0equiv y-frac{1}{4}pmod{35} Since 4^{-1}equiv 9pmod{35}, we have 0equiv y-9pmod{35}. Therefore, the y-intercept is (0,9). Finally, we have x_0+y_0=23+9=boxed{32}. The answer is: 32

question:What is the difference in the number of digits between the base-3 representation of 987_{10} and the base-8 representation of 987_{10}?

answer:To find the number of digits in a base-b representation of a positive integer n, we take the base-b logarithm of n and round up to the nearest integer. So the number of digits in the base-3 representation of 987_{10} is lceil log_3 987 rceil, and the number of digits in the base-8 representation of 987_{10} is lceil log_8 987 rceil. We have log_3 987 = frac{log_{10} 987}{log_{10} 3} approx frac{2.994}{0.477} approx 6.271, so the number of digits in the base-3 representation is 7. Similarly, log_8 987 = frac{log_{10} 987}{log_{10} 8} approx frac{2.994}{0.903} approx 3.319, so the number of digits in the base-8 representation is 4. Therefore, the difference in the number of digits is 7-4=boxed{3}. The answer is: 3

question:What is the simplified form of sqrt{245}?

answer:We can factor 245 as 245 = 49 cdot 5 = 7^2 cdot 5. Taking the square root, we have sqrt{245} = sqrt{7^2 cdot 5} = sqrt{7^2} cdot sqrt{5} = 7 sqrt{5}. So the simplified form of sqrt{245} is boxed{7sqrt{5}}.The answer is: 7sqrt{5}

question:Tabitha adds 1 serving of honey per cup of tea in the evening. She usually has 2 cups of tea before bed. She buys her honey in a 16-ounce container. If there are x servings of honey per ounce, She will be able to enjoy honey in her tea before bed for 48 nights. What is the value of unknown variable x? What is the value of unknown variable x?

answer:To solve this problem, we need to determine the value of x, which represents the number of servings of honey per ounce. Let's break down the information given: Number of servings of honey per cup of tea: 1 Number of cups of tea before bed: 2 Size of honey container: 16 ounces Number of nights honey will last: 48 We can set up the equation as follows: Number of servings of honey per cup of tea * Number of cups of tea before bed * Number of nights honey will last = Total number of servings of honey 1 * 2 * 48 = 16 * x Let's simplify and solve for x: 2 * 48 = 16x 96 = 16x To isolate x, we divide both sides of the equation by 16: 96 / 16 = 16x / 16 6 = x The value of x is 6. 6 The answer is: 6