Before, more or less 20 Years ago, we found such a conversation between a son and a father in the Parameter magazine. — Syn: "Daddy, today is the first day of the new year, and at the same time my birthday and yours. Do you know, Daddy: the sum of the digits of the new year is equal to this, how many years i am today, and last year it was not like that. Have you ever had such a coincidence??” - Father (after consideration): "No, such a coincidence did not happen to me ". — Syn: “And what year were you born in??” - Father: "If you like puzzles, I'll just tell you, that the sum of the digits of my birth year is divided by 9 ".
What year was the father born and the son born? When was this conversation?
Here is the solution:
Let us denote the year n, in which such a coincidence occurred for the first time, that the age of the son was equal to the sum of the digits of the number n. If we subtract the age of the son from the number n, then we will get the year of his birth. But the difference between any number and the sum of its digits is always divisible by 9. Hence the conclusion, that the son's year of birth is divided by 9. It wasn't a year 1935 years 1926, because for those years the coincidence only happened in a year 1950, relatively 1940. So it was a year 1917, and the coincidence happened in a year 1930. This coincidence repeated for a whole decade, up to a year 1939.
Not every birth year is divisible by 9 prejudges, that there will be a coincidence of age with the sum of the digits of the calendar year; the father was indeed born in a year divisible by 9, but such a coincidence did not happen to him. There was only one such year in the nineteenth century, namely 1881. If a man was born in 1881, it's up to a year 1899 his age was constantly less than the sum of the digits of the year calendarowego, a year 1900 - constantly bigger.
Father was born in 1881, son was born in the year 1917. The conversation took place on the day 1. I. 1930 r.