answer:For divisibility by 4, the number formed by the last two digits of the integer needs to be divisible by 4. The last two digits here form the number 5B. To determine if 5B is divisible by 4, B must yield a two-digit number divisible by 4 when placed at tens place. We examine the values from 0 to 9 for B: - When B = 0, 50 is divisible by 4. - When B = 1, 51 is not divisible by 4. - When B = 2, 52 is divisible by 4. - When B = 3, 53 is not divisible by 4. - When B = 4, 54 is divisible by 4. - When B = 5, 55 is not divisible by 4. - When B = 6, 56 is divisible by 4. - When B = 7, 57 is not divisible by 4. - When B = 8, 58 is divisible by 4. - When B = 9, 59 is not divisible by 4. The eligible candidates, therefore, are B = 0, 2, 4, 6, and 8. The largest digit among them that makes 5B divisible by 4 is B = 8. Thus, the largest suitable value for B is boxed{8}.

answer:Before the sale, the store had: Siamese cats: 20 Persian cats: 12 House cats: 8 Total number of cats before the sale = 20 + 12 + 8 = 40 During the sale, they sold: Siamese cats: 8 Persian cats: 5 House cats: 3 Total number of cats sold = 8 + 5 + 3 = 16 Total number of cats left after the sale = Total number of cats before the sale - Total number of cats sold Total number of cats left after the sale = 40 - 16 = 24 Therefore, there are boxed{24} cats left in the store after the sale.

answer:First, we calculate the area of each trapezoidal roof side: [ text{Area of trapezoid} = frac{1}{2} times (text{Base}_1 + text{Base}_2) times text{Height} = frac{1}{2} times (8 + 4) times 5 = 30 text{ square feet} ] Since there are two sides: [ text{Total area of roof} = 2 times 30 = 60 text{ square feet} ] Next, the area of the front face (equilateral triangle): [ text{Area of triangle} = frac{sqrt{3}}{4} times text{side}^2 = frac{sqrt{3}}{4} times 6^2 = 9sqrt{3} text{ square feet} ] The total area to be painted is: [ text{Total area} = 60 + 9sqrt{3} text{ square feet} ] The paint covers 100 square feet per gallon, so the gallons of paint needed: [ text{Gallons needed} = frac{60 + 9sqrt{3}}{100} approx frac{75.5}{100} = 0.755 text{ gallons} ] Rounding up to the nearest whole number, 1 gallon of paint is needed. The cost for the paint: [ text{Cost} = 1 times 15 = boxed{15} ]

answer:**Step 1:** Recall the trigonometric identity for the tangent of the sum of two angles: tan(A + B) = frac{tan A + tan B}{1 - tan A tan B} **Step 2:** Given that tan 60^circ = tan(20^circ + 40^circ), apply the trigonometric identity to expand the tangent of the sum: sqrt{3} = tan 60^circ = frac{tan 20^circ + tan 40^circ}{1 - tan 20^circ tan 40^circ} **Step 3:** Rearrange the equation to isolate the expression we are looking for: sqrt{3} - sqrt{3}tan 20^circ tan 40^circ = tan 20^circ + tan 40^circ **Step 4:** Add sqrt{3}tan 20^circ tan 40^circ to both sides of the equation: tan 20^circ + tan 40^circ + sqrt{3}tan 20^circ tan 40^circ = sqrt{3} Therefore, the value of tan 20^circ + tan 40^circ + sqrt{3}tan 20^circ tan 40^circ is boxed{sqrt{3}}.