Prickly Prime Problem
Number Theory
Level
pending
Two distinct two-digit prime numbers less than 60 are chosen. When their sum is subtracted from their product, which of the following values is impossible to attain?
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1 solution
Let the two primes be x and y .
The expression we have is x y − ( x + y ) = x y − x − y = ( x − 1 ) ( y − 1 ) − 1
We know that both x and y are odd, being primes (if either are two, so ( x − 1 ) ( y − 1 ) is divisible by 4 .
We therefore want a number n such that n + 1 ISN'T a multiple of four. It's clear that the answer is 1 1 3 7 as 1 1 3 7 + 1 = 1 1 3 8 isn't divisible by 4