Find the triplets

Number Theory Level 4

Let $a,b$ and $c$ be three integers that follows an arithmetic progression , and the product $a\times b \times c$ is a prime number .

How many ordered triplets of $(a,b,c)$ are possible?

The answer is 2.

2 solutions

Navneet Prabhat
Oct 13, 2016

-3,-1,1 &1,-1,-3 are the two solutions -1,1,3 is not a solution because -3 is not a prime. Prime numbers are defined only for natural numbers

Nice question.Do you have any rigorous proof that only 2 such ordered triplets are possible with given conditions?

Anandmay Patel - 4 years, 8 months ago

I have the proof Since a, b, c are integers therefore two of them must be either 1 or -1. Now you can search the third one by the fact that these three numbers are in AP. If you liked the question then please like the question and follow me for more problems.

Navneet Prabhat - 4 years, 8 months ago

@Navneet Prabhat @Tapas Mazumdar Thanks

Anandmay Patel - 4 years, 8 months ago

Check out this problem for the proof.

Tapas Mazumdar - 4 years, 8 months ago

Why is it not 6? I found these ordered pairs:

$(1,-1,-3) \\ (-1,1,-3) \\ (1,-3,-1) \\ (-1,-3,1) \\ (-3,1,-1) \\ (-3,-1,1)$

As you can see, we have three distinct integers for each ordered triplet, so number of ways to arrange them is $3! = 6$ .

Tapas Mazumdar - 4 years, 8 months ago

I'm sorry I got it. I didn't notice that you were referring to ordered triplets $(a,b,c)$ where $a, \ b$ and $c$ follow an AP in that sequence only.

Tapas Mazumdar - 4 years, 8 months ago

Siva Bathula
Oct 12, 2016

The answer is 4. ( -3, -1, 1) (1,-1,-3)(-1,1,3)(3,1,-1)

Find the triplets

The answer is 2.

2 solutions

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