💡Okay, so I have this problem where I need to find the value of 'a' such that two given lines are perpendicular. The lines are:First line: y = -3x - 7Second line: 9y + ax = 15Hmm, I remember that for two lines to be perpendicular, the product of their slopes should be -1. So, I need to find the slopes of these two lines and then set up an equation where their product equals -1.Starting with the first line, y = -3x - 7. This is already in slope-intercept form (y = mx + b), where m is the slope. So, the slope of the first line is -3. That part seems straightforward.Now, the second line is given by 9y + ax = 15. I need to find its slope. To do that, I should rewrite this equation in slope-intercept form as well. Let me try that.Starting with 9y + ax = 15.I want to solve for y, so I'll subtract ax from both sides:9y = -ax + 15Now, divide every term by 9 to get y by itself:y = (-a/9)x + 15/9Simplifying 15/9, which reduces to 5/3, so the equation becomes:y = (-a/9)x + 5/3Okay, so now the second line is in slope-intercept form, and its slope is -a/9.Now, since the two lines need to be perpendicular, the product of their slopes should be -1. So, let's set up that equation.Slope of first line * Slope of second line = -1Substituting the slopes we found:(-3) * (-a/9) = -1Let me compute the left side first. Multiplying -3 and -a/9:(-3) * (-a/9) = (3a)/9Simplifying 3a/9, which reduces to a/3.So, now the equation is:a/3 = -1To solve for 'a', I'll multiply both sides by 3:a = -1 * 3a = -3Wait, that seems straightforward, but let me double-check my steps to make sure I didn't make any mistakes.1. First line's slope is -3. Correct.2. Second line: 9y + ax = 15. Solving for y:9y = -ax + 15y = (-a/9)x + 5/3. So, slope is -a/9. Correct.3. Perpendicular condition: (-3) * (-a/9) = -1Calculating left side: (-3)*(-a/9) = 3a/9 = a/3. So, a/3 = -1Multiply both sides by 3: a = -3. That seems right.Wait a second, let me think again. If the product of the slopes is -1, then:m1 * m2 = -1Here, m1 is -3, m2 is -a/9.So, (-3)*(-a/9) = -1Which is 3a/9 = -1Simplify 3a/9 to a/3, so a/3 = -1Therefore, a = -3.Yes, that seems consistent. So, the value of 'a' should be -3.Just to make sure, let me plug a = -3 back into the second equation and see if the lines are indeed perpendicular.Second equation becomes: 9y + (-3)x = 15Which is 9y - 3x = 15Solving for y:9y = 3x + 15y = (3x + 15)/9Simplify:y = (1/3)x + 5/3So, the slope is 1/3.Now, the first line has a slope of -3, and the second line has a slope of 1/3.Multiplying them: (-3)*(1/3) = -1Which satisfies the condition for perpendicularity. So, yes, a = -3 is correct.I think that's solid. I don't see any mistakes in my calculations or reasoning. So, the answer should be a = -3.

💡First, I'll denote the total number of employees before July 1 as ( E ) and the average salary per employee as ( S ). Therefore, the total salary paid to all employees before July 1 is ( E times S ).On July 1, the number of employees decreased by 10%, so the new number of employees is 90% of ( E ), which is ( 0.9E ).The total salary paid to all employees after July 1 is said to be 100% of that before July 1, which means it is still ( E times S ).Now, the average salary per employee after July 1, which I'll call ( S' ), is the total salary divided by the new number of employees, which is ( frac{E times S}{0.9E} ).[S' = frac{E times S}{0.9E} = frac{S}{0.9} = S times left(frac{10}{9}right)]To find the percent increase in the average salary, I'll compare the new average salary ( S' ) to the original average salary ( S ).[text{Percent Increase} = left[frac{S' - S}{S}right] times 100%]Substituting ( S' ) with ( S times left(frac{10}{9}right) ):[text{Percent Increase} = left[frac{S times left(frac{10}{9}right) - S}{S}right] times 100%][text{Percent Increase} = left[frac{10S}{9} - S}{S}right] times 100%][text{Percent Increase} = left[frac{10S - 9S}{9S}right] times 100%][text{Percent Increase} = left[frac{S}{9S}right] times 100%][text{Percent Increase} = left(frac{1}{9}right) times 100%][text{Percent Increase} = 11.11%]Therefore, the average employee salary was increased by approximately ( boxed{11.11%} ).

💡Okay, so I have this problem here with four consecutive even numbers, a, b, c, d, where each is bigger than the last, and they're all greater than 2. I need to figure out which of the given expressions (A) through (E) will give me the largest value. The expressions are all fractions involving these numbers in different combinations. First, I should probably understand what consecutive even numbers mean. Consecutive even numbers are numbers that come one after another in the sequence of even numbers. For example, 4, 6, 8, 10 are four consecutive even numbers. So, if a is the first one, then b would be a + 2, c would be a + 4, and d would be a + 6. That makes sense because each is 2 more than the previous.Since the problem says 2 < a < b < c < d, I know that a has to be at least 4 because it's the first even number after 2. So, let me pick a specific set of numbers to make this concrete. Let's say a = 4, then b = 6, c = 8, and d = 10. I can use these numbers to plug into each of the expressions and see which one gives the largest value.Alright, let's go through each option one by one.**Option (A): (b + c)/(a + d)**Plugging in the numbers: (6 + 8)/(4 + 10) = 14/14 = 1. Okay, so that's 1.**Option (B): (a + d)/(c + b)**Again, plugging in: (4 + 10)/(8 + 6) = 14/14 = 1. Hmm, also 1.**Option (C): (c + d)/(b + a)**Substituting: (8 + 10)/(6 + 4) = 18/10 = 1.8. That's higher than 1.**Option (D): (a + c)/(b + d)**Plugging in: (4 + 8)/(6 + 10) = 12/16 = 0.75. That's less than 1.**Option (E): (b + d)/(c + a)**Substituting: (6 + 10)/(8 + 4) = 16/12 ≈ 1.333. That's about 1.33.So, comparing all the results:- A: 1- B: 1- C: 1.8- D: 0.75- E: ~1.33Clearly, 1.8 is the largest value here. So, option (C) gives the largest value.But wait, let me make sure I didn't make a mistake. Maybe I should try another set of consecutive even numbers to see if the pattern holds. Let's pick a = 6, then b = 8, c = 10, d = 12.**Option (A): (8 + 10)/(6 + 12) = 18/18 = 1****Option (B): (6 + 12)/(10 + 8) = 18/18 = 1****Option (C): (10 + 12)/(8 + 6) = 22/14 ≈ 1.57****Option (D): (6 + 10)/(8 + 12) = 16/20 = 0.8****Option (E): (8 + 12)/(10 + 6) = 20/16 = 1.25**Wait, in this case, option (C) is approximately 1.57, which is still the largest, but not as high as 1.8 in the previous set. Hmm, so maybe the first set I chose gave a higher value for (C). Let me try another set to see.Let's take a = 8, so b = 10, c = 12, d = 14.**Option (A): (10 + 12)/(8 + 14) = 22/22 = 1****Option (B): (8 + 14)/(12 + 10) = 22/22 = 1****Option (C): (12 + 14)/(10 + 8) = 26/18 ≈ 1.444****Option (D): (8 + 12)/(10 + 14) = 20/24 ≈ 0.833****Option (E): (10 + 14)/(12 + 8) = 24/20 = 1.2**Again, option (C) is the largest, but it's decreasing as I increase the starting number. So, in the first case, with a = 4, (C) was 1.8, which is the highest. In the next two cases, it's lower but still the highest among the options.Maybe the first set is the best case for (C). Let me check with a = 2, but wait, the problem says 2 < a, so a must be greater than 2. So, a = 4 is the smallest possible.Alternatively, maybe I can represent the numbers algebraically instead of plugging in specific numbers. Let me try that approach.Since a, b, c, d are consecutive even numbers, I can express them as:a = ab = a + 2c = a + 4d = a + 6Now, let's express each option in terms of a.**Option (A): (b + c)/(a + d) = ( (a + 2) + (a + 4) ) / (a + (a + 6)) = (2a + 6)/(2a + 6) = 1**So, regardless of the value of a, option (A) is always 1.**Option (B): (a + d)/(c + b) = (a + (a + 6))/( (a + 4) + (a + 2) ) = (2a + 6)/(2a + 6) = 1**Similarly, option (B) is always 1.**Option (C): (c + d)/(b + a) = ( (a + 4) + (a + 6) ) / ( (a + 2) + a ) = (2a + 10)/(2a + 2) = (2a + 10)/(2a + 2)**Let me simplify this:(2a + 10)/(2a + 2) = [2(a + 5)]/[2(a + 1)] = (a + 5)/(a + 1)So, this simplifies to (a + 5)/(a + 1). Let's see how this behaves as a increases.As a increases, both numerator and denominator increase, but the numerator is always 4 more than the denominator. So, (a + 5)/(a + 1) = 1 + 4/(a + 1). As a increases, 4/(a + 1) decreases, so the value approaches 1 from above.For the smallest a, which is 4, we get (4 + 5)/(4 + 1) = 9/5 = 1.8, which matches our earlier calculation.**Option (D): (a + c)/(b + d) = (a + (a + 4))/( (a + 2) + (a + 6) ) = (2a + 4)/(2a + 8) = (2(a + 2))/(2(a + 4)) = (a + 2)/(a + 4)**This simplifies to (a + 2)/(a + 4). Let's see:(a + 2)/(a + 4) = 1 - 2/(a + 4). As a increases, this approaches 1 from below.For a = 4, it's (6)/(8) = 0.75, which is what we saw earlier.**Option (E): (b + d)/(c + a) = ( (a + 2) + (a + 6) ) / ( (a + 4) + a ) = (2a + 8)/(2a + 4) = [2(a + 4)]/[2(a + 2)] = (a + 4)/(a + 2)**Simplifying, (a + 4)/(a + 2) = 1 + 2/(a + 2). As a increases, this approaches 1 from above.For a = 4, it's (8)/(6) ≈ 1.333, which is consistent with our earlier result.So, summarizing the algebraic results:- A and B are always 1.- C is (a + 5)/(a + 1), which is greater than 1 and decreases as a increases.- D is (a + 2)/(a + 4), which is less than 1 and increases as a increases.- E is (a + 4)/(a + 2), which is greater than 1 and decreases as a increases.Therefore, the maximum value among these occurs when a is as small as possible, which is a = 4. At a = 4, C is 1.8, E is approximately 1.333, and A and B are 1. So, 1.8 is the largest.To double-check, let's see if there's any other way this could be approached without plugging in numbers or algebra. Maybe by comparing the fractions directly.Looking at option (C): (c + d)/(b + a). Since c and d are the two largest numbers, their sum is going to be larger compared to the sum of b and a, which are the two smaller numbers. So, intuitively, this fraction should be larger than 1.Similarly, option (E): (b + d)/(c + a). Here, b and d are the second and fourth numbers, while c and a are the third and first. So, it's a mix, but since d is larger than c, and b is larger than a, it's not immediately clear which sum is bigger. However, when we calculated, it turned out to be about 1.333, which is less than 1.8.Options (A) and (B) both simplify to 1 because the numerator and denominator are the same when you add the numbers in a certain way.Option (D) is less than 1 because the numerator is smaller than the denominator.So, putting it all together, option (C) gives the largest value when a is the smallest possible even number greater than 2, which is 4. Therefore, the answer should be option (C).

💡Okay, so I have this problem to evaluate the expression: (3 + 1)(3² + 1²)(3⁴ + 1⁴). Hmm, that looks a bit complicated at first glance, but maybe I can break it down step by step. Let me think about how to approach this.First, I notice that each part of the product is a sum of powers of 3 and 1. The first term is (3 + 1), which is straightforward. The next term is (3² + 1²), and the last one is (3⁴ + 1⁴). It seems like each subsequent term is squaring the previous exponents. Maybe there's a pattern or a formula I can use here.I remember that sometimes when dealing with products of sums of squares, there's a way to simplify them using a telescoping product or some identity. Let me try to recall if there's a specific identity that applies here. I think there's something related to the difference of squares, but this is a sum, not a difference. Wait, maybe I can manipulate it somehow.Oh, right! If I multiply and divide by (3 - 1), which is 2, I can create a difference of squares in the first step. Let me try that:So, I can write the original expression as:(3 + 1)(3² + 1²)(3⁴ + 1⁴) = [(3 - 1)(3 + 1)(3² + 1²)(3⁴ + 1⁴)] / (3 - 1)Since (3 - 1) is 2, this becomes:= [ (3² - 1²)(3² + 1²)(3⁴ + 1⁴) ] / 2Now, simplifying (3² - 1²) using the difference of squares formula:= [ (9 - 1)(9 + 1)(81 + 1) ] / 2Calculating each term:9 - 1 = 89 + 1 = 1081 + 1 = 82So, substituting back:= (8 * 10 * 82) / 2First, multiply 8 and 10:8 * 10 = 80Then, multiply 80 by 82:80 * 82 = 6560Finally, divide by 2:6560 / 2 = 3280Wait, that seems straightforward. But let me double-check each step to make sure I didn't make a mistake.Starting again, the original expression is (3 + 1)(3² + 1²)(3⁴ + 1⁴). I decided to multiply and divide by (3 - 1) to create a telescoping product. That step seems correct because multiplying by (3 - 1)/(3 - 1) is just multiplying by 1, which doesn't change the value.Then, I used the difference of squares on (3 - 1)(3 + 1) to get (3² - 1²), which is 8. That's correct because 3² is 9 and 1² is 1, so 9 - 1 is 8.Next, I had (3² + 1²) which is 9 + 1 = 10, and (3⁴ + 1⁴) which is 81 + 1 = 82. Multiplying these together: 8 * 10 = 80, and 80 * 82 = 6560. Dividing by 2 gives 3280.Hmm, that seems consistent. Let me try calculating the original expression without the trick to see if I get the same result.Calculating each term separately:(3 + 1) = 4(3² + 1²) = 9 + 1 = 10(3⁴ + 1⁴) = 81 + 1 = 82Now, multiplying them together: 4 * 10 = 40, and 40 * 82 = 3280. Yep, that's the same result. So, both methods give me 3280, which makes me confident that the answer is correct.I think the key insight here was recognizing that multiplying by (3 - 1) allows me to use the difference of squares and simplify the expression step by step. It's a clever trick that avoids having to compute large exponents directly. I remember learning something about telescoping products where each step simplifies the expression, making the multiplication easier.Another way to think about it is that each multiplication by (3^n + 1^n) is effectively squaring the previous term and adding 1, which relates to the exponents doubling each time. So, starting from (3 + 1), multiplying by (3² + 1²) gives a term that can be simplified using the difference of squares, and so on.It's interesting how these algebraic identities can make seemingly complex expressions manageable. I should remember this technique for future problems where I encounter products of sums of powers. It might save me a lot of time instead of computing each term individually, especially with larger exponents.In summary, by strategically multiplying and dividing by (3 - 1), I was able to simplify the expression step by step using the difference of squares, leading me to the final answer of 3280. It's a neat application of algebraic identities to make calculations more efficient.

💡Okay, so I have this problem here where I need to find the smallest natural number ( a ) that satisfies two conditions related to a sequence defined by a recurrence relation. The sequence starts with ( a_0 = 0 ) and ( a_1 = 1 ), and each subsequent term is given by ( a_{n+1} = 2a_n + (a - 1)a_{n-1} ). The conditions are about prime numbers: for primes ( p ) less than or equal to a given prime ( p_0 ), ( p ) should divide ( a_p ), and for primes ( p ) greater than ( p_0 ), ( p ) should divide ( a_p ) as well. First, I need to understand the sequence better. It's a linear recurrence relation, so maybe I can solve it explicitly. The general form of a linear recurrence relation like this is usually solved by finding the characteristic equation. The characteristic equation for this recurrence should be ( r^2 - 2r - (a - 1) = 0 ). Let me write that down:( r^2 - 2r - (a - 1) = 0 )Solving this quadratic equation, the roots will be:( r = frac{2 pm sqrt{4 + 4(a - 1)}}{2} = frac{2 pm sqrt{4a}}{2} = 1 pm sqrt{a} )So, the general solution of the recurrence relation should be:( a_n = C(1 + sqrt{a})^n + D(1 - sqrt{a})^n )Now, applying the initial conditions to find constants ( C ) and ( D ). For ( n = 0 ):( a_0 = 0 = C(1 + sqrt{a})^0 + D(1 - sqrt{a})^0 = C + D )So, ( C + D = 0 ) which implies ( D = -C ).For ( n = 1 ):( a_1 = 1 = C(1 + sqrt{a}) + D(1 - sqrt{a}) )Substituting ( D = -C ):( 1 = C(1 + sqrt{a}) - C(1 - sqrt{a}) = C[ (1 + sqrt{a}) - (1 - sqrt{a}) ] = C(2sqrt{a}) )So, ( C = frac{1}{2sqrt{a}} ) and ( D = -frac{1}{2sqrt{a}} ).Thus, the explicit formula for ( a_n ) is:( a_n = frac{1}{2sqrt{a}} left[ (1 + sqrt{a})^n - (1 - sqrt{a})^n right] )Okay, so that's the general term. Now, I need to analyze this for prime numbers ( p ). The problem states two conditions:1. If ( p ) is a prime number and ( p leq p_0 ), then ( p ) divides ( a_p ).2. If ( p ) is a prime number and ( p > p_0 ), then ( p ) divides ( a_p ).Wait, both conditions say that ( p ) divides ( a_p ). Is that right? Let me check the original problem statement again.Oh, no, the second condition says ( p times a_p ). Hmm, that might mean ( p ) divides ( a_p ) as well, but maybe it's a typo or something. Maybe it's supposed to say ( p ) divides ( a_p ) for primes greater than ( p_0 ) as well. Or perhaps it's a different condition. Let me read it again."If ( p ) is a prime number and ( p > p_0 ), then ( p times a_p )." Hmm, that doesn't make much sense. Maybe it's supposed to say ( p ) divides ( a_p ) or ( a_p ) is divisible by ( p ). Maybe it's a translation issue or a typo. I'll assume that both conditions are that ( p ) divides ( a_p ) for primes ( p leq p_0 ) and for primes ( p > p_0 ). So, essentially, for all primes ( p ), ( p ) divides ( a_p ). But that might not make sense because if ( a ) is fixed, ( a_p ) would have to be divisible by all primes, which is impossible unless ( a_p = 0 ), but ( a_p ) is non-zero for ( p geq 1 ). So, maybe I misinterpret the second condition.Wait, the original problem says:"(ii) If ( p ) is a prime number and ( p > p_0 ), then ( p times a_{p} )."Hmm, that's not a complete statement. It should probably say ( p ) divides ( a_p ) or ( a_p ) is divisible by ( p ). Maybe in the original Chinese, it was clear, but in translation, it's cut off. Alternatively, maybe it's saying ( p ) does not divide ( a_p ), but that would contradict the first condition. Hmm.Alternatively, maybe it's saying that ( p ) divides ( a_p ) for ( p leq p_0 ), and ( p ) does not divide ( a_p ) for ( p > p_0 ). That would make more sense, as it's asking for the smallest ( a ) such that primes up to ( p_0 ) divide ( a_p ), but primes above ( p_0 ) do not. That seems like a more reasonable problem.So, assuming that, the problem is:(i) For primes ( p leq p_0 ), ( p ) divides ( a_p ).(ii) For primes ( p > p_0 ), ( p ) does not divide ( a_p ).So, we need the smallest ( a ) such that ( a_p equiv 0 mod p ) for primes ( p leq p_0 ), and ( a_p notequiv 0 mod p ) for primes ( p > p_0 ).Alright, that makes more sense. So, now, with that understanding, let's proceed.Given the explicit formula for ( a_p ):( a_p = frac{1}{2sqrt{a}} left[ (1 + sqrt{a})^p - (1 - sqrt{a})^p right] )We can analyze this modulo ( p ). Let's consider ( a_p mod p ). Since ( p ) is prime, we can use Fermat's little theorem or properties of binomial coefficients modulo ( p ).First, expand ( (1 + sqrt{a})^p ) and ( (1 - sqrt{a})^p ) using the binomial theorem:( (1 + sqrt{a})^p = sum_{k=0}^p binom{p}{k} (sqrt{a})^k )Similarly,( (1 - sqrt{a})^p = sum_{k=0}^p binom{p}{k} (-1)^k (sqrt{a})^k )Subtracting these two:( (1 + sqrt{a})^p - (1 - sqrt{a})^p = 2 sum_{k=0}^{lfloor p/2 rfloor} binom{p}{2k+1} (sqrt{a})^{2k+1} )So, all the even terms cancel out, and we're left with twice the sum of the odd terms. Therefore,( a_p = frac{1}{2sqrt{a}} times 2 sum_{k=0}^{lfloor p/2 rfloor} binom{p}{2k+1} (sqrt{a})^{2k+1} )Simplify:( a_p = sum_{k=0}^{lfloor p/2 rfloor} binom{p}{2k+1} a^{k} )Now, modulo ( p ), we can note that ( binom{p}{k} equiv 0 mod p ) for ( 1 leq k leq p-1 ). So, all the binomial coefficients except for ( binom{p}{0} ) and ( binom{p}{p} ) are divisible by ( p ). However, in our sum, the binomial coefficients are ( binom{p}{2k+1} ), which for ( 1 leq 2k+1 leq p-1 ) are divisible by ( p ). The only term that isn't necessarily divisible by ( p ) is when ( 2k+1 = p ), but since ( p ) is odd (as it's a prime greater than 2), ( 2k+1 = p ) implies ( k = frac{p-1}{2} ). So, the term when ( k = frac{p-1}{2} ) is ( binom{p}{p} a^{(p-1)/2} = 1 times a^{(p-1)/2} ).Therefore, modulo ( p ), all terms except the last one vanish, and we have:( a_p equiv a^{(p-1)/2} mod p )So, ( a_p equiv a^{(p-1)/2} mod p ).Therefore, the condition ( p mid a_p ) is equivalent to ( a^{(p-1)/2} equiv 0 mod p ). But ( a ) is a natural number, so ( a^{(p-1)/2} equiv 0 mod p ) implies that ( a ) must be divisible by ( p ). Because if ( a ) is not divisible by ( p ), then ( a^{(p-1)/2} ) modulo ( p ) is either 1 or -1, depending on whether ( a ) is a quadratic residue modulo ( p ) or not. But since we need ( a^{(p-1)/2} equiv 0 mod p ), ( a ) must be divisible by ( p ).Therefore, for each prime ( p leq p_0 ), ( p ) must divide ( a ). So, ( a ) must be a multiple of all primes ( p leq p_0 ). Now, for primes ( p > p_0 ), we need ( p nmid a_p ). From the above, ( a_p equiv a^{(p-1)/2} mod p ). So, ( a_p notequiv 0 mod p ) implies that ( a^{(p-1)/2} notequiv 0 mod p ), which means ( a ) is not divisible by ( p ). Therefore, ( a ) must not be divisible by any prime ( p > p_0 ).Therefore, ( a ) must be the product of all primes ( p leq p_0 ). Since we need the smallest such ( a ), it's the product of all primes up to ( p_0 ).Wait, but let me check this. If ( a ) is the product of all primes up to ( p_0 ), then for primes ( p leq p_0 ), ( p ) divides ( a ), so ( a_p equiv 0 mod p ). For primes ( p > p_0 ), since ( a ) is not divisible by ( p ), ( a^{(p-1)/2} ) modulo ( p ) is either 1 or -1, so ( a_p equiv pm 1 mod p ), which means ( p nmid a_p ). Therefore, the smallest such ( a ) is the product of all primes up to ( p_0 ). But wait, let me test this with a small example. Let's take ( p_0 = 3 ). Then, the primes up to ( p_0 ) are 2 and 3. So, ( a = 2 times 3 = 6 ). Let's compute ( a_p ) for ( p = 2, 3, 5 ).For ( p = 2 ):( a_2 = 2a_1 + (a - 1)a_0 = 2 times 1 + 5 times 0 = 2 ). So, ( 2 mid 2 ), which is good.For ( p = 3 ):( a_3 = 2a_2 + 5a_1 = 2 times 2 + 5 times 1 = 4 + 5 = 9 ). So, ( 3 mid 9 ), which is good.For ( p = 5 ):We need to compute ( a_5 ). Let's compute the sequence up to ( a_5 ):( a_0 = 0 )( a_1 = 1 )( a_2 = 2 times 1 + 5 times 0 = 2 )( a_3 = 2 times 2 + 5 times 1 = 4 + 5 = 9 )( a_4 = 2 times 9 + 5 times 2 = 18 + 10 = 28 )( a_5 = 2 times 28 + 5 times 9 = 56 + 45 = 101 )Now, check if 5 divides 101. 101 divided by 5 is 20.2, so no, 5 does not divide 101. So, that's good because ( p = 5 > p_0 = 3 ), and indeed, 5 does not divide ( a_5 ).So, this seems to work for ( p_0 = 3 ). Let's try another example with ( p_0 = 5 ). Then, ( a ) should be the product of primes up to 5, which is ( 2 times 3 times 5 = 30 ).Compute ( a_p ) for ( p = 2, 3, 5, 7 ).For ( p = 2 ):( a_2 = 2a_1 + 29a_0 = 2 times 1 + 29 times 0 = 2 ). So, 2 divides 2.For ( p = 3 ):( a_3 = 2a_2 + 29a_1 = 2 times 2 + 29 times 1 = 4 + 29 = 33 ). 3 divides 33.For ( p = 5 ):Compute up to ( a_5 ):( a_0 = 0 )( a_1 = 1 )( a_2 = 2 )( a_3 = 33 )( a_4 = 2 times 33 + 29 times 2 = 66 + 58 = 124 )( a_5 = 2 times 124 + 29 times 33 = 248 + 957 = 1205 )Check if 5 divides 1205. 1205 divided by 5 is 241, so yes, 5 divides 1205.For ( p = 7 ):Compute ( a_7 ):( a_6 = 2 times 1205 + 29 times 124 = 2410 + 3596 = 6006 )( a_7 = 2 times 6006 + 29 times 1205 = 12012 + 34945 = 46957 )Check if 7 divides 46957. 46957 divided by 7 is approximately 6708.142..., so no, 7 does not divide 46957. So, that's good.Therefore, this seems to hold. So, the smallest ( a ) is indeed the product of all primes up to ( p_0 ).But wait, let me think again. Is there a smaller ( a ) that satisfies the conditions? For example, maybe ( a ) doesn't have to be the product of all primes up to ( p_0 ), but just needs to be divisible by each prime up to ( p_0 ). So, the minimal ( a ) would be the least common multiple (LCM) of the primes up to ( p_0 ). But since primes are coprime, the LCM is just their product. So, yes, the minimal ( a ) is the product of all primes up to ( p_0 ).Therefore, the answer should be the product of all primes less than or equal to ( p_0 ).**Final Answer**The smallest value of ( a ) is the product of all primes up to ( p_0 ), which is boxed{prod_{text{primes } p leq p_0} p}.

💡Alright, so I have this geometry problem here: two circles intersect at points A and B, and MN is their common tangent. I need to prove that the line AB bisects the segment MN. Hmm, okay, let me try to visualize this.First, I imagine two circles overlapping, intersecting at points A and B. The common tangent MN touches both circles at points M and N, respectively. So, MN is a line that just touches each circle at one point without crossing it. Now, I need to show that the line connecting the two intersection points, AB, cuts the tangent MN exactly in half.Let me recall some geometry concepts that might help here. I remember that the tangent to a circle is perpendicular to the radius at the point of tangency. So, if I draw radii from the centers of each circle to points M and N, those radii should be perpendicular to MN.Wait, but the problem doesn't mention the centers of the circles. Maybe I need to consider the centers? Let me denote the centers of the two circles as O1 and O2. So, O1 is the center of the first circle, and O2 is the center of the second circle. Then, O1M and O2N are radii perpendicular to MN.Now, since MN is a common tangent, the lengths from M to O1 and from N to O2 are equal because they're both radii. But I'm not sure if that's directly useful here.Another thought: maybe I can use the power of a point. The power of a point with respect to a circle is equal to the square of the length of the tangent from that point to the circle. So, if I take point O, where AB intersects MN, then the power of O with respect to both circles should be equal because O lies on AB, which is the radical axis of the two circles.Wait, the radical axis! That's the line AB itself, right? The radical axis of two circles is the set of points that have equal power with respect to both circles. So, any point on AB has equal power with respect to both circles. That means the power of point O with respect to both circles is the same.Since MN is a common tangent, the power of point O with respect to each circle is equal to the square of the length of the tangent from O to each circle. But since MN is the common tangent, the tangent lengths from O to each circle should be equal. Therefore, OM equals ON, meaning O is the midpoint of MN.Wait, let me make sure I'm not skipping any steps here. So, if O is on AB, the radical axis, then the power of O with respect to both circles is equal. The power of O with respect to the first circle is OM squared, and with respect to the second circle is ON squared. Therefore, OM squared equals ON squared, which implies OM equals ON. Hence, O is the midpoint of MN.That seems to make sense. But let me try to think if there's another way to approach this, maybe using similar triangles or something else.If I draw the radii O1M and O2N, both perpendicular to MN, and since MN is a common tangent, O1M and O2N are both equal in length because they're radii. Also, the line O1O2 connecting the centers is perpendicular to MN because both radii are perpendicular to MN.Wait, no, actually, O1O2 is not necessarily perpendicular to MN unless the circles are congruent. Hmm, maybe that's not the right approach.Alternatively, maybe I can consider triangles formed by the centers and the tangent points. Let's see, triangle O1MO and triangle O2NO. Since O1M and O2N are both radii and perpendicular to MN, these triangles are right triangles. If I can show that these triangles are congruent, then OM would equal ON.But for triangles O1MO and O2NO to be congruent, we need more information. We know O1M equals O2N (radii), and both are right angles. But we don't know if O1O2 is equal to O2O1 or anything about the other sides. Maybe this isn't the best approach.Going back to the power of a point, that seems more straightforward. Since O is on the radical axis AB, its power with respect to both circles is equal, so OM squared equals ON squared, hence OM equals ON. Therefore, O is the midpoint of MN, meaning AB bisects MN.I think that's a solid argument. Maybe I should write it out more formally to make sure I haven't missed anything.So, to summarize:1. Let O be the intersection point of AB and MN.2. Since O lies on AB, the radical axis of the two circles, the power of O with respect to both circles is equal.3. The power of O with respect to each circle is equal to the square of the length of the tangent from O to that circle.4. Since MN is a common tangent, the tangent lengths from O to each circle are OM and ON.5. Therefore, OM squared equals ON squared, implying OM equals ON.6. Hence, O is the midpoint of MN, meaning AB bisects MN.Yeah, that seems to cover it. I don't think I've missed any steps here. The key was recognizing that O lies on the radical axis, which ensures equal power with respect to both circles, leading to equal tangent lengths and thus O being the midpoint.