answer:1. **Express (a^4 - b^4) in a factored form:** [ a^4 - b^4 = (a^2)^2 - (b^2)^2 = (a^2 - b^2)(a^2 + b^2) ] Further factorize (a^2 - b^2): [ a^2 - b^2 = (a - b)(a + b) ] Therefore: [ a^4 - b^4 = (a - b)(a + b)(a^2 + b^2) ] 2. **Check divisibility by 5:** Since (a) and (b) are primes greater than 5, they are not divisible by 5. By Fermat's Little Theorem, for any integer (p) (where (p) is a prime): [ p^4 equiv 1 pmod{5} ] Therefore: [ a^4 equiv 1 pmod{5} quad text{and} quad b^4 equiv 1 pmod{5} ] Thus: [ a^4 - b^4 equiv 1 - 1 equiv 0 pmod{5} ] This shows (a^4 - b^4) is divisible by 5. 3. **Check divisibility by 3:** Similarly, by Fermat's Little Theorem for primes (a) and (b) greater than 3: [ a^4 equiv 1 pmod{3} quad text{and} quad b^4 equiv 1 pmod{3} ] Thus: [ a^4 - b^4 equiv 1 - 1 equiv 0 pmod{3} ] This shows (a^4 - b^4) is divisible by 3. 4. **Check divisibility by 16:** Since (a) and (b) are odd primes, (a equiv 1 pmod{2}) and (b equiv 1 pmod{2}). Therefore: [ a^2 equiv 1 pmod{8} quad text{and} quad b^2 equiv 1 pmod{8} ] Thus: [ a^4 equiv 1 pmod{16} quad text{and} quad b^4 equiv 1 pmod{16} ] Therefore: [ a^4 - b^4 equiv 1 - 1 equiv 0 pmod{16} ] This shows (a^4 - b^4) is divisible by 16. 5. **Combine the results:** Since (a^4 - b^4) is divisible by 5, 3, and 16, it is divisible by their least common multiple: [ text{lcm}(5, 3, 16) = 240 ] Therefore: [ 240 mid (a^4 - b^4) ] 6. **Show that 240 is the greatest positive integer with this property:** Consider the example given in the problem: [ a = 13, quad b = 11 quad Rightarrow quad a^4 - b^4 = 13^4 - 11^4 = 28561 - 14641 = 13920 = 240 cdot 58 ] [ a = 17, quad b = 13 quad Rightarrow quad a^4 - b^4 = 17^4 - 13^4 = 83521 - 28561 = 54960 = 240 cdot 229 ] In both cases, 240 is a factor, and no larger common factor exists that divides all such differences for any primes (a) and (b). (blacksquare)

answer:Let the radius of the circle be r units. The area of the circle is given by the formula A = pi r^2. Given that the area A = 4pi, we have: [ pi r^2 = 4pi ] Dividing both sides of the equation by pi, we get: [ r^2 = 4 ] Taking the square root of both sides: [ r = 2 ] The diameter d of the circle is twice the radius: [ d = 2r = 2 times 2 = 4 ] Thus, the diameter of the circle is boxed{4} units.

answer:[Analysis] This problem primarily tests the knowledge of vector coordinate operations in a plane. The key is to find the coordinates of vectors a and b. After that, use the formula cos theta = dfrac{a cdot b}{|a||b|} to obtain the solution. [Step-by-step Solution] 1. Find the coordinates of vectors a and b. From the given equations, we have: a + b = (5, -10) a - b = (3, 6) Adding these equations, we get: 2a = (5+3, -10+6) 2a = (8, -4) a = (4, -2) Subtracting the second equation from the first, we get: 2b = (5-3, -10-6) 2b = (2, -16) b = (1, -8) 2. Using the formula for the cosine of the angle between two vectors, we have: cos theta = dfrac{a cdot b}{|a||b|} Where: - a cdot b is the dot product of vectors a and b. - |a| and |b| are the magnitudes of vectors a and b, respectively. Calculate the dot product: a cdot b = (4, -2) cdot (1, -8) = (4)(1) + (-2)(-8) = 4 + 16 = 20 Calculate the magnitudes of a and b: |a| = sqrt{(4)^2 + (-2)^2} = sqrt{16 + 4} = sqrt{20} = 2sqrt{5} |b| = sqrt{(1)^2 + (-8)^2} = sqrt{1 + 64} = sqrt{65} Substitute these values into the cosine formula: cos theta = dfrac{20}{(2sqrt{5})(sqrt{65})} = dfrac{20}{2sqrt{5 cdot 65}} = dfrac{20}{2sqrt{5 cdot 13 cdot 5}} = boxed{dfrac{2sqrt{13}}{13}}

answer:First, let's convert the time it takes to groom the animals from hours to minutes, since the total time given is in minutes. 1 hour = 60 minutes For dogs: 2.5 hours/dog * 60 minutes/hour = 150 minutes/dog For cats: 0.5 hours/cat * 60 minutes/hour = 30 minutes/cat Now, let's calculate the time it takes to groom 3 cats: 3 cats * 30 minutes/cat = 90 minutes Since the total time to groom the animals is 840 minutes, we can subtract the time it took to groom the cats to find out how much time was spent grooming dogs: 840 minutes - 90 minutes = 750 minutes Now, we can calculate the number of dogs groomed by dividing the time spent grooming dogs by the time it takes to groom one dog: 750 minutes / 150 minutes/dog = 5 dogs Therefore, boxed{5} dogs were groomed.