Prime Numbers Math Question

My DD was asked this question in one of the paper, can you please explain how to solve it,

Two different prime numbers between 4 and 18 are chosen. When their sum is subtracted from their product, which of the following number could be obtained?

(1) 21

(2) 60

(3) 119

(4) 180

(5) 231

Great question Kevin,

Here is the way to solve such questions:

Let the two prime numbers be x and y.

Question is : xy - (x+y) = ? or xy -x -y +1 -1 (adding 1 and -1 = 0)
Or in another words

=> (1-x) * (1-y) -1

=>Since x and y both are prime numbers (x-1) and (y-1) is going to be an even number.

=>x-1 and y-1 product is also going to be an even number

=> Even * Even is always divisible by 4, so the question is asking which number from all the options when divided by 4 gives remainder 3 (as there is a -1 in [(1-x)*(1-y) -1]), the only option is 119 as its one short of 120 and is completely divisible by 4.

So the answer is 119. (3)

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@ClassRx thanks for the explanation,

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