Evaristo's grandfather

Evaristo's grandfather

Evaristo's grandfather, who is not a centenary, is 63 years older than Evaristo, who is of legal age. The sum of the ages of the grandfather and Evaristo is equal to the sum of the figures of their ages multiplied by a certain integer n and the difference of the ages is equal to the same number n multiplied by the difference between the figures of the age of the grandfather and the sum of Evaristo's age figures,

What is the age of Evaristo's grandfather?


Grandpa is 84 years old and Evaristo 21. (n = 7)

Actually, the second fact about relationships between ages is enough to solve the problem.

Being the difference between both ages equal to 63, which is a multiple of 9, the difference between the figures of one and another age, by congruence with 9, is equal to 9 or 0. Therefore, as the product of that difference by n is 63 (difference between ages), n cannot be more than 7. This immediately leads the solution (84,21) as the respective ages.