Agustín, Benito, Carlos, Diego, Esteban and Federico are six collectors of paintings of which two of them are brothers. One day they went to an exhibition together and bought as follows: Agustin bought

1 painting, Benito bought 2, Carlos 3, Diego 4, Esteban 5 and Federico 6 paintings.

The two brothers paid the same amount of money for each of the paintings they bought. The rest of the people in the group paid double for each picture of what the brothers paid for each of their own. In total, the six collectors spent € 100,000. We know that the price of each painting was a whole number of euros.

**Who are the brothers?**

#### Solution

In total we have 21 pictures. Let's call * Y* to the number of paintings the brothers buy,

*to the number of paintings that other collectors buy and*

**z***to what it costs the brothers each picture.*

**x**So we have that xy + 2xz = 100,000 and y + z = 21. Where we get that x (y + 2z) = 100,000 and therefore x (21 + z) = 100,000.

z could be 10, 11, 12, 13, 14, 15, 16, 17 and 18 (all possible combinations of sums of 4 of the six quantities of frames purchased: 1, 2, 3, 4, 5 and 6) but * z* it can only be 11, since by adding

*to 21 has to give a divisor of 100,000.*

**z**To be * z*=11,

*it will be 10. The only two that can add up to 10 frames are the ones that bought 4 and 6 so*

**Y****the brothers are therefore Diego and Federico**.