💡To find the largest integer less than 100 that leaves a remainder of 5 when divided by 8, I'll start by identifying the largest multiple of 8 that is less than 100.First, I divide 100 by 8, which gives 12 with a remainder of 4. This means the largest multiple of 8 less than 100 is 12 times 8, which equals 96.Next, I add the desired remainder of 5 to this multiple: 96 plus 5 equals 101. However, 101 is greater than 100, so I need to find the next lower multiple of 8.The next lower multiple of 8 is 11 times 8, which equals 88. Adding the remainder of 5 to this gives 88 plus 5 equals 93.Finally, I verify that 93 is less than 100 and that when divided by 8, it leaves a remainder of 5. Both conditions are satisfied, so 93 is the correct answer.

💡First, I need to determine the remainder when 964,807 is divided by 13. This will help me understand how much needs to be subtracted to make the number divisible by 13.I'll start by performing the division of 964,807 by 13. 964,807 ÷ 13 equals 74,215 with a remainder of 2. This means that 964,807 is 2 more than a multiple of 13. To make the number divisible by 13, I need to subtract this remainder from the original number.Therefore, subtracting 2 from 964,807 will result in a number that is exactly divisible by 13.

💡Okay, so I have this problem where I need to find the area of a parallelogram constructed on two vectors, a and b. The vectors are given in terms of p and q, which are other vectors. The problem also gives me the magnitudes of p and q, and the angle between them. Hmm, let's see.First, I remember that the area of a parallelogram formed by two vectors is equal to the magnitude of their cross product. So, if I can find the cross product of vectors a and b, and then find its magnitude, that should give me the area. That makes sense because the cross product gives a vector perpendicular to both a and b, and its magnitude is equal to the area of the parallelogram.Alright, so let me write down the given vectors:a = 3p - 2qb = p + 5qAnd the magnitudes:|p| = 4|q| = 1/2The angle between p and q is 5π/6. Okay, so that's 150 degrees. I might need the sine of this angle for the cross product.Now, I need to compute the cross product of a and b. Let me recall how cross products work. The cross product of two vectors is distributive, so I can expand it like regular multiplication.So, a × b = (3p - 2q) × (p + 5q)Let me expand this:= 3p × p + 3p × 5q - 2q × p - 2q × 5qWait, let me make sure I distribute correctly. So, it's 3p × p + 3p × 5q - 2q × p - 2q × 5q.Simplify each term:First term: 3p × p. But I remember that the cross product of any vector with itself is zero. So, p × p = 0. Therefore, 3p × p = 0.Second term: 3p × 5q. That's 15(p × q).Third term: -2q × p. Hmm, cross product is anti-commutative, meaning q × p = -p × q. So, -2q × p = -2(-p × q) = 2(p × q).Fourth term: -2q × 5q. Again, q × q = 0, so this term is zero.So, putting it all together:a × b = 0 + 15(p × q) + 2(p × q) + 0Combine like terms:= (15 + 2)(p × q)= 17(p × q)Okay, so the cross product a × b is 17 times (p × q). Now, I need the magnitude of this cross product.The magnitude of a cross product |a × b| is equal to |a||b|sin(theta), where theta is the angle between a and b. But in this case, since a × b is 17(p × q), the magnitude will be 17 times |p × q|.So, |a × b| = 17|p × q|Now, I need to find |p × q|. Using the formula for the magnitude of the cross product:|p × q| = |p||q|sin(theta)Where theta is the angle between p and q, which is given as 5π/6.So, let's compute |p × q|:|p| = 4|q| = 1/2sin(5π/6) = sin(π - π/6) = sin(π/6) = 1/2So, |p × q| = 4 * (1/2) * (1/2) = 4 * 1/4 = 1Therefore, |a × b| = 17 * 1 = 17So, the area of the parallelogram is 17.Wait, let me double-check my steps to make sure I didn't make any mistakes.1. Expanded the cross product correctly: yes, used distributive property.2. Recognized that p × p and q × q are zero: correct.3. Handled the anti-commutative property for q × p: yes, turned it into p × q with a positive sign.4. Combined coefficients correctly: 15 + 2 = 17: correct.5. Calculated |p × q|: used the formula, substituted the values correctly, sin(5π/6) is indeed 1/2.6. Multiplied everything correctly: 4 * 1/2 * 1/2 = 1, then 17 * 1 = 17.Everything seems to check out. I think the area is indeed 17.**Final Answer**The area of the parallelogram is boxed{17}.

💡First, I need to determine how many files Megan had left after deleting 21 of them. She originally had 93 files.Next, I'll subtract the 21 deleted files from the total to find out how many files remain: 93 - 21 = 72 files.Now, Megan wants to organize these remaining files into folders, with each folder containing 8 files. To find out how many folders she needs, I'll divide the number of remaining files by the number of files per folder: 72 ÷ 8 = 9.Therefore, Megan ended up with 9 folders.

💡First, I need to understand the diagram provided. It shows a circle with several lines drawn from the center, creating different angles labeled with expressions involving ( x ).I notice that the angles around the center of the circle are ( 4x^circ ), ( 5x^circ ), ( 2x^circ ), and ( x^circ ). Since the sum of all angles around a point is ( 360^circ ), I can set up an equation to find the value of ( x ).Adding up all the angles:[4x + 5x + 2x + x = 360]Combining like terms:[12x = 360]Solving for ( x ):[x = frac{360}{12} = 30]Therefore, the value of ( x ) is ( 30 ).

💡First, I need to determine the number of men and women on the ferry. Since 40% of the 1500 passengers are men, I can calculate the number of men by multiplying 1500 by 0.40. This gives me 600 men. The remaining passengers are women, so I subtract the number of men from the total passengers to find there are 900 women.Next, I'll calculate how many women are wearing hats. If 15% of the women are wearing hats, I multiply 900 by 0.15, which equals 135 women wearing hats.Similarly, I'll determine the number of men wearing hats. With 12% of the men wearing hats, I multiply 600 by 0.12, resulting in 72 men wearing hats.Finally, to find the total number of passengers wearing hats, I add the number of women wearing hats to the number of men wearing hats: 135 + 72 equals 207 passengers in total wearing hats.