Question

Три брата родились в один и тот же день, но в разные года. Когда старшему из них исполнилось 17 лет, то сумма возрастов всех трех братьев нацело разделилась на 17. Докажите, что когда среднему из братьев исполнится 17 лет, то сумма возрастов всех братьев не будет кратной 17.

Answer 1

Очевидно что сумма их возрастов не превышает 34, так как если     старшему 17 , то сумма среднего и младшего не должно быть больше 17.
то есть пусть их возраста равняются n и 17-n 
тогда когда старшему будет 17 , пусть через х лет то 
$17+x+n+x+(17-n+x)=34+3x$ очевидно не кратна 17 

Answer 2

Problem Analysis

We are given that three brothers were born on the same day but in different years. When the oldest brother turned 17, the sum of the ages of all three brothers was divisible by 17. We need to prove that when the middle brother turns 17, the sum of the ages of all three brothers will not be divisible by 17.

Proof

Let's assume the ages of the three brothers are A, B, and C, where A is the age of the oldest brother, B is the age of the middle brother, and C is the age of the youngest brother.

According to the given information, when the oldest brother turned 17, the sum of their ages was divisible by 17. Mathematically, we can represent this as:

(A + B + C) % 17 = 0

Now, we need to prove that when the middle brother turns 17, the sum of their ages will not be divisible by 17. Mathematically, we need to prove:

(A + B + C) % 17 ≠ 0

To prove this, we can use a proof by contradiction. Let's assume that when the middle brother turns 17, the sum of their ages is still divisible by 17:

(A + B + C) % 17 = 0

Now, let's consider the difference in ages between the oldest and middle brothers, and the middle and youngest brothers:

Oldest - Middle = A - B Middle - Youngest = B - C

Since the brothers were born on the same day, the difference in their ages will always be an integer. Therefore, we can represent the difference in ages as:

(A - B) % 17 = X (B - C) % 17 = Y

Now, let's substitute the values of A - B and B - C in the equation (A + B + C) % 17 = 0:

(A + B + C) % 17 = (A - B + B - C + C) % 17 = (A - C) % 17 = (X + Y) % 17

Since X and Y are integers, (X + Y) % 17 will also be an integer. Therefore, if (A + B + C) % 17 = 0, then (X + Y) % 17 = 0.

However, this contradicts the given information that when the oldest brother turned 17, the sum of their ages was divisible by 17. Therefore, our assumption that (A + B + C) % 17 = 0 when the middle brother turns 17 is false.

Hence, we have proved that when the middle brother turns 17, the sum of their ages will not be divisible by 17.

Conclusion

We have proven that when the middle brother turns 17, the sum of the ages of all three brothers will not be divisible by 17.