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If two integers are chosen at random out of the set {2, 5, 7, 8}, what

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Joined: 16 Feb 2018
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Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what  [#permalink]

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New post 29 Sep 2018, 06:20
IanStewart wrote:
You can avoid an exhaustive test here. Suppose I ask whether (97)(103) can be written in the form a^2 - b^2, where a and b are integers. Notice that this is a difference of squares: a^2 - b^2 = (a+b)(a-b). We can now just use the median of 97 and 103, which is 100:

(97)(103) = (100-3)(100+3) = 100^2 - 3^2

So whenever we can write our product in such a way that the median of our two numbers is an integer, we can write our product as a difference of squares just as above. For example, if we take 5*7, that's equal to (6-1)(6+1), and if we take 2*8, that's equal to (5-3)(5+3). Now if we look at 8*5, we can't immediately use the same trick, but we can 'move' one of the 2s from the 8 into the 5, as follows: 8*5 = 4*10 = (7-3)(7+3). Similarly, 8*7 = 4*14 = (9-5)(9+5). So of our six possible products, four can be written as a difference of squares.


I am sorry, i still dont know how you got the answer from this. What to do after these steps.
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Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what  [#permalink]

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New post 01 Oct 2018, 04:35
lostaish wrote:
IanStewart wrote:
You can avoid an exhaustive test here. Suppose I ask whether (97)(103) can be written in the form a^2 - b^2, where a and b are integers. Notice that this is a difference of squares: a^2 - b^2 = (a+b)(a-b). We can now just use the median of 97 and 103, which is 100:

(97)(103) = (100-3)(100+3) = 100^2 - 3^2

So whenever we can write our product in such a way that the median of our two numbers is an integer, we can write our product as a difference of squares just as above. For example, if we take 5*7, that's equal to (6-1)(6+1), and if we take 2*8, that's equal to (5-3)(5+3). Now if we look at 8*5, we can't immediately use the same trick, but we can 'move' one of the 2s from the 8 into the 5, as follows: 8*5 = 4*10 = (7-3)(7+3). Similarly, 8*7 = 4*14 = (9-5)(9+5). So of our six possible products, four can be written as a difference of squares.


I am sorry, i still dont know how you got the answer from this. What to do after these steps.


Check out this post: https://www.veritasprep.com/blog/2014/0 ... at-part-i/

A number can be written in the form a^2 - b^2 if it is odd or has 4 as a factor (explained in the post above)

There are 4C2 = 6 ways of picking a pair of numbers here. Out of these, in only two cases (2, 5) and (2, 7) you will not be able to write the product as a^2 - b^2. Since product will not be odd and will not have 4 as a factor.
So required probability = 4/6 = 2/3
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Re: If two integers are chosen at random out of the set {2, 5, 7, 8}, what &nbs [#permalink] 01 Oct 2018, 04:35

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