three consecutive primes are summed. when the sum is squared, the result is 72,361. what is the largest of the three primes?
(sum of 3 consecutive primes )^2 = 72361
sum of 3 consecutive primes = 269
269/3 = 89
so the middle prime must be around 89
BUT 89 is a prime
http://primes.utm.edu/lists/small/10000.txt
the one before is 83 and the one after is 97
So the largest of these primes is 97
check:
(83+89+97)^
= 269^2
= 72361 , YEAH!
Well, well, well, let's play with these primes, shall we? So, we have three consecutive primes and their sum squared is 72,361. Let's do some math magic.
Now, to find the largest prime, we need to find the square root of 72,361, which is... *drumroll*... 269!
No, no, I'm not clowning around! 269 is indeed the square root of 72,361.
But wait, there's more! Since we have three consecutive primes, let's find the middle one. So, if we subtract 1 from 269, we get 268. And hey presto, 268 is an even number and therefore not a prime!
So, the largest prime would be the one before 268, which is... *cue suspenseful music*... 267!
Congratulations, 267 is the largest prime among the three! I hope you enjoyed this prime-time comedy.
Let's denote the three consecutive primes as p, p+2, and p+4. According to the given information, we have the equation:
(p + p+2 + p+4)^2 = 72,361
Expanding the left side of the equation, we get:
(3p + 6)^2 = 72,361
Simplifying further, we have:
9p^2 + 36p + 36 = 72,361
Subtracting 72,361 from both sides, we get:
9p^2 + 36p - 72,325 = 0
To solve this quadratic equation, we can use the quadratic formula:
p = (-b ± sqrt(b^2 - 4ac)) / 2a
In this case, a = 9, b = 36, and c = -72,325. Substituting these values into the quadratic formula, we get:
p = (-36 ± sqrt(36^2 - 4(9)(-72,325))) / 2(9)
Simplifying further, we have:
p = (-36 ± sqrt(1296 + 26,071)) / 18
p = (-36 ± sqrt(27,367)) / 18
To find the largest prime, we need to select the positive value of p. Evaluating the expression inside the square root, we get:
sqrt(27,367) ≈ 165.604
Therefore, we have:
p = (-36 + 165.604) / 18 ≈ 6.867
Since p must be a prime number, we can conclude that the largest prime is the nearest whole number to 6.867, which is 7.
Therefore, the largest of the three primes is 7.
To find the largest of the three consecutive primes, we need to start by breaking down the problem into steps:
Step 1: Find the square root of 72,361.
Step 2: Identify the three consecutive numbers that, when summed, add up to the number obtained in step 1.
Step 3: Determine which of these three numbers is the largest and is also a prime number.
Let's go through each step in detail:
Step 1: Finding the square root of 72,361
To find the square root of a number, we can use a calculator or a programming language. In this case, taking the square root of 72,361 gives us 269.
Step 2: Identifying the consecutive numbers that add up to 269.
We need to find three consecutive numbers that sum up to 269. Let's consider the middle number as 'x'. The numbers will be x-1, x, and x+1. Their sum is (x-1) + x + (x+1) = 3x. Thus, we need to find a number 'x' such that 3x equals 269.
Dividing 269 by 3 gives us approximately 89.67. Since 'x' must be an integer, it means that 'x' is rounded to the nearest whole number, which is 90. Therefore, the three numbers are 89, 90, and 91.
Step 3: Determining the largest prime number among the three.
To check if a number is prime, we need to see if it can only be divided evenly by 1 and itself, without any other factors. Let's check if 91 is prime.
We can find if a number is prime manually by checking all numbers less than or equal to the square root of the number. However, this process can be time-consuming for large numbers. Alternatively, we can use an algorithm such as the Sieve of Eratosthenes to determine prime numbers efficiently.
By applying the Sieve of Eratosthenes to 91, we can determine that it is not a prime number because it is divisible evenly by 7 and 13.
So, out of the three consecutive numbers, 89, 90, and 91, the largest prime is 89.
Therefore, the largest of the three primes is 89.