Question

Вовочка и Гогочка строят башни из кубиков. У Вовочки кубики деревянные, со стороной 27 мм. А у Гогочки пластмассовые кубики, со стороной 48 мм. После долгого кропотливого труда Вовочка и Гогочка исхитрились-таки построить башни одинаковой высоты, каждый из своих кубиков. Найдите наименьшую возможную высоту башни Гогочки (ну или Вовочки, если он вам более симпатичен) в миллиметрах.

Answer 1

Нам надо найти наименьшее общее кратное:
27 = 3 * 3 * 3 = 3^3;
48 = 2 * 2 * 2 * 3 = 2^3 *3;
нок = 2^3 * 3^3 = 8  * 27 = 216mm

Answer 2

Problem Analysis

Vovochka and Gogochka are building towers using wooden and plastic cubes, respectively. The wooden cubes have a side length of 27 mm, while the plastic cubes have a side length of 48 mm. They want to build towers of equal height using their own cubes. We need to find the minimum possible height of Gogochka's tower (or Vovochka's tower if preferred) in millimeters.

Solution

To find the minimum possible height of Gogochka's tower, we need to determine the greatest common divisor (GCD) of the side lengths of the wooden and plastic cubes. The GCD will represent the height of the tower that can be built using both types of cubes.

The GCD can be calculated using the Euclidean algorithm. The algorithm states that the GCD of two numbers can be found by repeatedly subtracting the smaller number from the larger number until the two numbers become equal. The resulting number is the GCD.

In this case, we need to find the GCD of 27 mm and 48 mm.

Calculation

To find the GCD of 27 mm and 48 mm, we can use the Euclidean algorithm:

1. Divide 48 mm by 27 mm: 48 mm ÷ 27 mm = 1 remainder 21 mm. 2. Divide 27 mm by 21 mm: 27 mm ÷ 21 mm = 1 remainder 6 mm. 3. Divide 21 mm by 6 mm: 21 mm ÷ 6 mm = 3 remainder 3 mm. 4. Divide 6 mm by 3 mm: 6 mm ÷ 3 mm = 2 remainder 0 mm.

Since the remainder is now 0, we can conclude that the GCD of 27 mm and 48 mm is 3 mm.

Answer

Therefore, the minimum possible height of Gogochka's tower (or Vovochka's tower) is 3 mm.

[[1]]