💡Okay, so I have this problem where Pinocchio has a bunch of 5 and 6 soldi coins, more than 10 of each. He buys a book that costs N soldi and can't pay exactly for it without needing change. I need to find the largest possible N that's no more than 50.Alright, let me start by understanding what it means to not be able to pay exactly without change. That means there's no combination of 5s and 6s that add up to N. So, I need to find the largest N ≤ 50 that can't be expressed as 5a + 6b, where a and b are non-negative integers.First, I should recall something about the Frobenius number, which is the largest number that can't be expressed as a combination of two coprime numbers. The formula for two coprime numbers x and y is xy - x - y. In this case, 5 and 6 are coprime because their greatest common divisor is 1.So, applying the formula, the Frobenius number would be 5*6 - 5 - 6 = 30 - 5 - 6 = 19. That suggests that 19 is the largest number that can't be expressed as a combination of 5s and 6s. But wait, the problem says N is not more than 50, and 19 is less than 50. So, does that mean 19 is the answer?But hold on, the Frobenius number applies when you have only two coin denominations, and they are coprime. Here, we have 5 and 6, which are coprime, so the formula should hold. But let me double-check to make sure.Let me list out the numbers from 1 to, say, 25 and see which ones can be expressed as combinations of 5 and 6.1: Can't be expressed.2: Can't be expressed.3: Can't be expressed.4: Can't be expressed.5: Yes, one 5.6: Yes, one 6.7: Can't be expressed.8: Can't be expressed.9: Can't be expressed.10: Yes, two 5s.11: Can't be expressed.12: Yes, two 6s.13: Can't be expressed.14: Can't be expressed.15: Yes, three 5s.16: Can't be expressed.17: Can't be expressed.18: Yes, three 6s.19: Can't be expressed.20: Yes, four 5s.21: Yes, three 5s and one 6.22: Yes, two 5s and two 6s.23: Yes, one 5 and three 6s.24: Yes, four 6s.25: Yes, five 5s.Okay, so up to 25, 19 is indeed the largest number that can't be expressed as a combination of 5s and 6s. But the problem says N is not more than 50. So, does that mean 19 is still the answer because beyond 19, all numbers can be expressed?Wait, let me check numbers beyond 25 to see if there are any numbers up to 50 that can't be expressed as combinations of 5s and 6s.26: Yes, four 5s and one 6.27: Yes, three 5s and two 6s.28: Yes, two 5s and three 6s.29: Yes, one 5 and four 6s.30: Yes, five 6s or six 5s.31: Yes, five 5s and one 6.32: Yes, four 5s and two 6s.33: Yes, three 5s and three 6s.34: Yes, two 5s and four 6s.35: Yes, seven 5s.36: Yes, six 6s.37: Yes, five 6s and one 5.38: Yes, four 6s and two 5s.39: Yes, three 6s and three 5s.40: Yes, eight 5s.41: Yes, seven 5s and one 6.42: Yes, six 6s.43: Yes, five 6s and two 5s.44: Yes, four 6s and three 5s.45: Yes, nine 5s.46: Yes, eight 5s and one 6.47: Yes, seven 5s and two 6s.48: Yes, six 6s and two 5s.49: Yes, five 6s and four 5s.50: Yes, ten 5s.Wow, so from 20 up to 50, every number can be expressed as a combination of 5s and 6s. That means the largest number that can't be expressed is indeed 19. So, even though the problem allows N up to 50, the answer is still 19 because beyond 19, all numbers can be paid without needing change.But just to be thorough, let me think about why the Frobenius number works here. Since 5 and 6 are coprime, the largest number that can't be expressed as a combination of them is 19. This is because once you pass 19, you can always add 5s or 6s to reach higher numbers. For example, if you can express 20, then adding 5s will get you to 25, 30, etc., and adding 6s will get you to 26, 32, etc. Since 5 and 6 are consecutive numbers after 19, they cover all the gaps.So, yeah, I think 19 is the correct answer. It's the largest number less than or equal to 50 that can't be expressed as a combination of 5s and 6s.

💡First, I need to determine how long it takes for the man to complete the work alone. I'll denote the number of days it takes for the man to do the work alone as ( M ) days.Next, I'll calculate the work rates of both the man and his son. The man's work rate is ( frac{1}{M} ) of the work per day, and the son's work rate is ( frac{1}{6} ) of the work per day since he can complete the work in 6 days.When they work together, their combined work rate is the sum of their individual work rates, which is ( frac{1}{M} + frac{1}{6} ) of the work per day.We know that together they can complete the work in 3 days, so their combined work rate is ( frac{1}{3} ) of the work per day.Setting up the equation:[frac{1}{M} + frac{1}{6} = frac{1}{3}]To solve for ( M ), I'll find a common denominator and simplify the equation:[frac{6}{6M} + frac{M}{6M} = frac{2M}{6M}][frac{6 + M}{6M} = frac{2M}{6M}]Since the denominators are the same, I'll equate the numerators:[6 + M = 2M]Subtracting ( M ) from both sides:[6 = M]Therefore, the man alone can complete the work in 6 days.

💡Okay, so I need to find the annual interest rate for a savings account where 5,000 was deposited and after 6 months, the amount grew to 5,202. The interest is compounded quarterly. Hmm, let's break this down step by step.First, I remember that compound interest can be calculated using the formula:[ A = P left(1 + frac{r}{n}right)^{nt} ]Where:- ( A ) is the amount of money accumulated after n years, including interest.- ( P ) is the principal amount (the initial amount of money).- ( r ) is the annual interest rate (in decimal form).- ( n ) is the number of times that interest is compounded per year.- ( t ) is the time the money is invested for, in years.In this problem:- ( P = 5,000 )- ( A = 5,202 )- ( n = 4 ) (since it's compounded quarterly)- ( t = frac{1}{2} ) year (because 6 months is half a year)So, plugging these values into the formula:[ 5202 = 5000 left(1 + frac{r}{4}right)^{4 times frac{1}{2}} ]Simplifying the exponent:[ 5202 = 5000 left(1 + frac{r}{4}right)^2 ]Now, I need to solve for ( r ). First, divide both sides by 5000 to isolate the term with ( r ):[ frac{5202}{5000} = left(1 + frac{r}{4}right)^2 ]Calculating the left side:[ 1.0404 = left(1 + frac{r}{4}right)^2 ]To get rid of the exponent, I'll take the square root of both sides:[ sqrt{1.0404} = 1 + frac{r}{4} ]Calculating the square root of 1.0404:[ 1.020098 approx 1 + frac{r}{4} ]Subtracting 1 from both sides:[ 0.020098 approx frac{r}{4} ]Now, multiply both sides by 4 to solve for ( r ):[ r approx 0.080392 ]To express this as a percentage, I'll multiply by 100:[ r approx 8.0392% ]So, the annual interest rate is approximately 8.04%.Wait, let me double-check my calculations to make sure I didn't make any mistakes. Starting from the beginning, the formula seems correct. Plugging in the numbers, the division and square root steps look accurate. The final multiplication by 4 and conversion to a percentage also seem right. Yeah, I think this is correct.

💡First, identify the largest five-digit number, which is 99999.Next, divide 99999 by 8 to find how many times 8 fits into it.The division yields a quotient of 12499.875.Since we need a whole number, take the integer part of the quotient, which is 12499.Multiply this integer quotient by 8 to find the largest multiple of 8 that is less than or equal to 99999.12499 multiplied by 8 equals 99992.Therefore, the largest five-digit number divisible by 8 is 99992.

💡First, I identify the coefficients of the quadratic equation (x^{2} - 5x + 2 = 0). Here, (a = 1), (b = -5), and (c = 2).Next, I recall the formula for the discriminant of a quadratic equation, which is (Delta = b^{2} - 4ac).Substituting the identified values into the formula:[Delta = (-5)^{2} - 4 times 1 times 2 = 25 - 8 = 17]Therefore, the discriminant of the quadratic equation is (17).

💡First, I need to determine which of the given equations is incorrect.I'll start by evaluating each option one by one.For option A: 8 + 34 = 32. Adding 8 and 34 should give 42, not 32. This indicates that option A is incorrect.To ensure accuracy, I'll verify the other options as well.Option B: 17 + 17 = 34. Adding 17 and 17 indeed results in 34, so this is correct.Option C: 15 + 13 = 28. Adding 15 and 13 gives 28, which is correct.Since option A is the only one that does not hold true, it is the incorrect equation.