Are you joking?

Number Theory Level 4

Jack said:"I'm 37 and my wife is 32; we have got 3 sons: Luisa who is 11, Carl, who is 7 and Alex who is......... If you would like to know Alex's age, it's enough to elevate my age to the exponent 'age of my mother-in-law', divide the result by Luisa's age and see the remainder. But that's not all: you can also elevate my wife's age to the 'age of the waitress',divide the result by Carl's age and see the remainder. In both cases the result is the same: Alex's age. However, while you solve the problem, i will go to supermarket to buy a candle for Alex's cake. I'll add it to the candles of last year".

So, which is Alex's age??

The answer is 4.

1 solution

Seth Lovelace
Nov 29, 2014

${ 37 }^{ X }\quad mod\quad 11\\ { 4 }^{ 0 }\equiv 1\\ { 4 }^{ 1 }\equiv 4\\ { 4 }^{ 2 }\equiv 5\\ { 4 }^{ 3 }\equiv 9\\ { 4 }^{ 4 }\equiv 3$

${ 32 }^{ Y }\quad mod\quad 7\\ { 4 }^{ 0 }\equiv 1\\ { 4 }^{ 1 }\equiv 4\\ { 4 }^{ 2 }\equiv 2$

The two equations can be derived from the information given, as can the cycle lengths that follow. Because 4 is the only common remainder between the two equations, and, "In both cases the result [remainder] is the same: Pierino's age," Pierino's age must be $\boxed{4}$ .

If you wish to learn more, check out one of Brilliant's notes or Wiki pages on mods, specifically exponential modular arithmetic in the case of this problem.

Just wanted to add something: 1 is also a common remainder. But, since there was a candle last year, this means that Pierino was not 0 years old last year. This leaves us with the answer of 4. :)

Neil Chua Goy - 6 years, 6 months ago

Nice problem by the way! I like the way you crafted it all together so nicely. @DropTheProblem

Seth Lovelace - 6 years, 6 months ago

It has numerous solutions though.

Mohamed Ramdan - 6 years ago

Are you joking?

The answer is 4.

1 solution

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