How Many Animals Does Bert Have?

  • Thread starter K Sengupta
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In summary, a "pet and prime problem" is a mathematical puzzle that involves rearranging the digits of a number to find the largest prime number possible. To solve it, the digits must be arranged in a specific order and checked for primality. An example would be rearranging 6259 to create 9659, the largest prime number. There are limitations, such as the given number must have at least two digits and cannot be a perfect square. Solving pet and prime problems can have practical applications in areas such as cryptography and computer programming, and can also help improve critical thinking and problem-solving skills.
  • #1
K Sengupta
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Bert has some cows, horses and dogs, a different prime number of each.

If the number of cows (c) is multiplied by the total of cows and horses (c+h), the product is 120 more than the number of dogs (d), that is:
c*(c+h) = 120 + d.

How many cows, horses and dogs does Bert have?
 
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  • #2
K Sengupta said:
Bert has some cows, horses and dogs, a different prime number of each.

If the number of cows (c) is multiplied by the total of cows and horses (c+h), the product is 120 more than the number of dogs (d), that is:
c*(c+h) = 120 + d.

How many cows, horses and dogs does Bert have?
I can't think of any "formula" for solving something like this but "guess and check" is an old, respected method for solving problems. 120 is about 11 squared so start by trying numbers around that. If c= 11, h= 3 c+h= 13, c(c+h)= 11(13)= 143= 120+ 23 and 23 is prime! c= 11, h= 3, d= 23 works.
 
  • #3
Of course, h=even and c=even is impossible.

If h=2 , c=odd
c**2 + 2c = 120 + d
(c+1)**2 = 11**2 + d , so d is not prime.

If h=odd , c=odd, then d=2 , and c*(c+h) = 2*61 , which is impossible.

If c=2 and h=odd, then d=2. Then, c=2 and h=59.

:smile:

Edited:

Of course, I should have noted "c" and "d" cannot be 2 ( Remember: "a different prime number").
And then, I should have seen d=(c+12)(c-10) is prime if c=11, which implies d=23, and h=2.
 
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  • #4
Rogerio said:
(c+1)**2 = 11**2 + d , so d is not prime.
This is incorrect. It yields,
d = (c-10)(c+12)
which is prime if c=11. So h=2, c=11, d =23 is a solution (I assume this is the one HallsofIvy meant).
 
  • #5
I have edited my post so I can pretend I didn't make that mistake!
 
  • #6
What about 2 cows, 59 horses and 2 dogs?
 
  • #7
Interestingly, no more solutions for primes < 100000.
 
  • #8
Borek said:
What about 2 cows, 59 horses and 2 dogs?

Borek, the question says that each of the three primes is different.

Bert has some cows, horses and dogs, a different prime number of each.
 
  • #9
Ah OK, somehow missed this condition.
 

Related to How Many Animals Does Bert Have?

1. What is a "pet and prime problem"?

A pet and prime problem is a mathematical puzzle that involves using the digits of a number to find the largest prime number possible. The "pet" in the name refers to the digits being rearranged in a specific way, while the "prime" refers to finding the largest prime number.

2. How do you solve a "pet and prime problem"?

To solve a pet and prime problem, you must first rearrange the digits of the given number in a specific order. The digits must be arranged in descending order from left to right, with any duplicates removed. Then, starting from the largest number created, check if it is a prime number. If it is not, move on to the next largest number until you find the largest prime number possible.

3. What is an example of a "pet and prime problem"?

For example, if the given number is 6259, it can be rearranged to create 9652, 9625, or 9659. Starting from the largest number, 9659 is a prime number and therefore the solution to the pet and prime problem for 6259.

4. Are there any limitations to solving a "pet and prime problem"?

Yes, there are a few limitations when it comes to solving a pet and prime problem. The given number must have at least two digits, and it cannot be a perfect square or have all of its digits the same. Additionally, the solution to a pet and prime problem may not always be unique, meaning there may be more than one possible answer.

5. What are the practical applications of solving a "pet and prime problem"?

While solving pet and prime problems may seem like a purely mathematical exercise, it can also have real-world applications. For example, it can be used in cryptography to create secure passwords or in computer programming to generate random numbers. It can also help improve critical thinking and problem-solving skills.

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