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# Prime number Expression riddle.

Thu 13 Jan, 2011 04:17 am
Hello,

(a * b) - c = d * e

where

a,b,c,d,e are unique prime numbers
a - b = c
b - d = c
c is smallest of the 5 numbers.

Can you guys guess the values of a,b,c,d,e? Is this solution unique?
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View best answer, chosen by vinsan
Oylok

1
Thu 13 Jan, 2011 05:59 am
@vinsan,
7,5,2,3,11
(yes, they're unique)
0 Replies

Oylok

1
Thu 13 Jan, 2011 06:13 am
@vinsan,
The justification is as follows:

a,b,d,e have to be odd, because they're larger than c (and only the smallest prime is even, namely 2)

c is the difference of two odds, so it must be even; it is prime, so it must be 2

the two simpler equations tell us that d, b, and a are three consecutive odd primes, so they must be 3,5,and 7 (above that point every third prime is divisible by 3)

that leaves (7*5) - 2 = 3 * e; so e = 11
Oylok

1
Fri 14 Jan, 2011 12:51 am
@Oylok,
Oops. I need to correct something here...

Oylok wrote:

a,b,d,e have to be odd, because they're larger than c (and only the smallest prime is even, namely 2)

c is the difference of two odds, so it must be even; it is prime, so it must be 2

the two simpler equations tell us that d, b, and a are three consecutive odd numbers that are all prime, so they must be 3,5,and 7 (above that point every third prime is divisible by 3)

that leaves (7*5) - 2 = 3 * e; so e = 11

That's better.
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vinsan

1
Mon 17 Jan, 2011 09:22 pm
But how can we prove its unique?
Oylok

1
Tue 18 Jan, 2011 05:20 am
@vinsan,
The justification I gave for my answer was the proof of its uniqueness. I mean, if I had just guessed the answer, there would have been no way of knowing whether my answer was unique, but since I deduced the answer as a logical consequence of the facts and equations you provided, it means there are no other possible answers.
Oylok

2
Tue 18 Jan, 2011 06:02 am
@Oylok,
I went over the proof and discovered something else that I typed incorrectly. It must be that posting proofs online makes me nervous.

Here is the uniqueness proof again, reworded slightly:

Step 1:
c is smallest, so none of the unknowns except c can possible be the smallest prime; therefore, a,b,d and e must all be odd, because only the smallest prime is even.

Step 2:
c must be even, because it is the difference between two odd numbers.

Step 3:
c must equal 2, because it is both even and prime.

Step 4:
The simpler equations tell us that d + 2 = b, and b + 2 = a; so d, b and a must be three consecutive odd numbers.

Step 5:
The only time you find three consecutive odd numbers that are all prime is with 3, 5 and 7. Above that level when you have three consecutive odds, one of them will always be composite (i.e. not prime), because it will be divisible by 3. So we know that a=7, b=5, and d=3.

Step 6:
This just leaves us with 7 * 5 - 2 = 3 * e. The only solution to that is e = 11.

Step 7:
I have just shown that if your conditions hold then the answer is "a=7, b=5, c=2, d=3, e=11." The contrapositive of that statement is, if the answer is not "a=7, b=5, c=2, d=3, e=11", then your conditions do not all hold.

In other words, "a=7, b=5, c=2, d=3, e=11" is the only acceptable answer. It is therefore the unique solution.
Oylok

1
Tue 18 Jan, 2011 06:08 am
@Oylok,
Oylok wrote:
posting proofs online makes me nervous

(That's the main reason why I post in maths threads, by the way; to overcome those nerves.)
0 Replies

engineer

2
Tue 18 Jan, 2011 08:53 am
@vinsan,
Oylok's reasoning shows why it is unique. Because C must be even and prime, it must be 2 with no other possible answers. Because three primes must be consecutive odd numbers and every third odd number is divisble by three, the numbers must be 3, 5 and 7. Those are all unique and that drives e to be uniquely eleven. Very nice reasoning.
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vinsan

2
Wed 19 Jan, 2011 10:37 pm
@Oylok,
@Oylok

Thx a lot for the reasoning. It becomes clearer at Step 5 onwards.

0 Replies

raprap

1
Mon 7 Feb, 2011 03:36 pm
@Oylok,
Nice and unique-the only prime triples within 4 of each other are 3, 5 & 7
0 Replies

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