Question 592549: the sum of the digits of a two-digit prime number is subtracted from the number. Prove that the difference cannot be a prime nuber. Answer by solver91311(24713) (Show Source):
The fact that the original number is prime is just a red herring. In fact, if you subtract the sum of the digits from any integer, the result is divisible by 9 and therefore not prime.
Proof for two-digit integers:
Let represent the 10s digit and let represent the 1s digit.
Then the original integer is and the sum of the digits is .
Subtracting:
Which is divisible by 9. So if ANY two digit integer less the sum of its digits is non-prime, any prime two digit integer must exhibit the same characteristic.
Q.E.D.
John
My calculator said it, I believe it, that settles it
