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Vandaag is het 16 January 2018

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Grandfather's birthday - math puzzle

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More than twenty years ago, Tim asked his grandfather who was celebrating his birthday that day: "How old are you now?" Grandfather smiled and said: "Add the digits of my year of birth and multiply the answer by 4." Tim would have looked so uncomfortable that his Grandfather immediately added, "By the way, this trick also works with the year we have on the calendar now." What was grandfathers age?

Explanation

If the sum of the digits of two numbers is the same, then the difference should be divisible by nine. For example 53 - 17 = 36, or 209 - 56 = 153. This is a general feature of numbers. The age of grandfather is hence a multiple of 9. Therefore the sum of the digits can in principle only be 9, 18, 27, .. But if grandfather was born in the 20th century the sum of the digits should above 10. So we are left with 18, 27, ... But if the sum would be 27 or larger grandfather should been born after 1989 and would be 108 years old which is clearly impossible. Hence the sum of the digits is 18 and grandfather is 4 * 18 = 72 years old. Grandfather was born in 1908.